Jobs and Unemployment in Macroeconomic
Theory: A Turbulence Laboratory
Thomas J. Sargent∗
Lars Ljungqvist
August 26, 2005
Abstract
We use three general equilibrium frameworks with jobs and unemployed workers
to study the effects of government mandated unemployment insurance (UI) and employment protection (EP). To illuminate the forces in these models, we study how UI
and EP affect outcomes when there is higher ‘turbulence’ in the sense of worse skill
transition probabilities for workers who suffer involuntary layoffs. Matching and searchisland models have labor market frictions and incomplete markets. The representative
family model with employment lotteries has no labor market frictions and complete
markets. The adverse welfare state dynamics coming from high UI indexed to past
earnings that were isolated by Ljungqvist and Sargent (1998) are so strong that they
determine outcomes in all three frameworks. Another force stressed by Ljungqvist and
Sargent (2005), through which higher layoff taxes suppress frictional unemployment in
less turbulent times, prevails in the models with labor market frictions, but not in the
frictionless representative family model. In addition, the high aggregate labor supply
elasticity that emerges from employment lotteries and complete insurance markets in
the representative family model makes it impossible to include generous governmentsupplied unemployment insurance in that model without getting the unrealistic result
that economic activity virtually shuts down.
Key words: Job, search, matching, lotteries, skills, turbulence, unemployment, unemployment insurance, employment protection, discouraged worker.
Ljungqvist: Stockholm School of Economics and CEPR (email: lars.ljungqvist@hhs.se); Sargent: New
York University and Hoover Institution (email: thomas.sargent@nyu.edu). We are thankful to Riccardo
Colacito, Constantino Hevia, Kevin Kallock, Sagiri Kitao, and Alejandro Rodriguez for excellent research
assistance. For useful comments on earlier drafts, we thank David Backus, Gadi Barlevy, Marco Bassetto,
Jeffrey Campbell, Dirk Krueger, Mariacristina De Nardi, Wouter DenHaan, David Domeij, Jesus FernandezVillaverde, Timothy Kehoe, Narayana Kocherlakota, Lisa Lynch, Edward C. Prescott, Richard Rogerson,
François Velde, and seminar participants at Georgetown University, New York University, the Federal Reserve
Banks of Chicago and New York, ITAM, the Stockholm School of Economics, the University of Helsinki,
the University of Oregon, and the European University Institute. Ljungqvist’s research was supported by a
grant from the Jan Wallander and Tom Hedelius Foundation. Sargent’s research was supported by a grant
to the National Bureau of Economic Research from the National Science Foundation.
∗
1
1
Introduction
Lucas (1987, p. 50) noted that real business cycle and other competitive equilibrium models
have “. . . no sense in which anyone . . . can be said to ‘have a job’ or lose, seek or find a job.”1
For analyzing positive and normative questions about unemployment compensation, Lucas
wanted a theory that includes reasons ‘why people allocate time to a particular activity –
like unemployment . . .’ (p. 54), and also why there are ‘jobs’ in the sense of long term
employee-employer relationships between particular workers and particular capitalists.
Macroeconomists disagree about the best models of jobs and the workers who do and
don’t have them. But we have several promising models. This paper compares three stochastic general equilibrium models containing jobs and uses them to study the effects of government supplied unemployment insurance (UI in the language of Mortensen and Pissarides
(1999)) and employment protection provisions (EP in the language of Mortensen and Pissarides (1999)). To illuminate the economic forces stressed within each model, we study how
increases in UI and EP interact with an increase in the probability of skill deterioration after
involuntary layoffs, where following Ljungqvist and Sargent (1998), we refer to an enlarged
skill deterioration probability as an increase in turbulence.
The models are (1) a suite of matching models inspired by Mortensen and Pissarides
(1999) and DenHaan et al. (2001); (2) a variant of a search-island model of Alvarez and
Veracierto (2001); and (3) an employment lotteries model with comprehensive economy-wide
insurance arrangements in the spirit of Hansen (1985), Rogerson (1988), and Prescott (2002).
The models share the same stochastic skill accumulation and deterioration technologies, but
differ with respect to labor market frictions and arrangements for allocating risks. Each
model has a theory of a job and of why unemployed workers choose to spend time in an
activity that can be called unemployment.2 In the matching models, workers without jobs
wait in a matching function. In the search-island model, depending on their financial assets,
human capital, and entitlement to benefits, some people spend more time unemployed than
others because they exert less effort searching. In the representative family lotteries model,
unemployed workers have won a lottery telling them to specialize in leisure.
1. The matching models have risk neutral workers, no asset accumulation decisions, an
attenuated allocative role for wages, and a significant allocative role for the ratio of
vacancies to unemployed workers.3 They feature adverse congestion effects that unemployed workers impose on each other and that firms with vacancies impose on each
1
Labor economists often use models in which there are no jobs. A useful example is the Rosen schooling
model that is estimated by Ryoo and Rosen (2004).
2
Because it has no theory of jobs, Lucas (1987, p. 53, footnote 4) indicated that he regarded the HansenRogerson model to be an inappropriate tool for analyzing policy questions concerning UI. We extend the
Hansen-Rogerson model to contain jobs. We include it among our suite of models partly because Prescott
(2002), Rogerson (2005a), and Rogerson (2005b, text for SED plenary lecture) have forcefully advocated it
as a vehicle for explaining the most pertinent European labor market outcomes, and also because we want
to determine whether the interaction between turbulence and UI that we featured in Ljungqvist and Sargent
(1998) also comes through in this environment.
3
Hosios (1990) describes the matching framework as follows. “Though wages in the matching-bargaining
2
other, a wage bargaining process, and waiting times as equilibrating signals that reconcile the decisions of firms and workers.
2. The search-island model has risk averse workers whose decisions about how intensively
to search when unemployed depend on their skills and benefit entitlements as well
as their accumulations of a risk-free asset, the only savings vehicle available to them.
The model features incomplete risk sharing through self-insurance against both unemployment and uncertain life spans in retirement. The wage rate is determined in a
competitive labor market but is constrained to remain fixed in the face of idiosyncratic
productivity shocks: workers receive a fixed wage per unit of skill for the duration of
a job. As a result, there are socially wasteful separations.
3. In the lotteries model, risk averse agents have access to complete markets in historycontingent consumption claims. Labor contracts are identical to those in the searchisland model. An indivisibility in the choice set of each worker requires him either to
work a fixed number of hours or not at all. That would confront the individual worker
with a nonconvexity in his opportunity set if he were isolated from other workers. But
he is not. He is a member of an economy-wide representative family that uses lotteries
to convexify its production opportunity set by assigning fractions of its members to
specialize in work and in leisure. The employment lottery yields a high aggregate labor
supply elasticity that makes the fraction of the national family that works respond
sensitively to both tax wedges and UI benefits.
These alternative labor market arrangements create different avenues through which UI and
EP influence outcomes, not just for unemployment, but for individual workers’ consumption
and labor market experiences including the incidence of long-term unemployment.
Our frameworks share some common limitations. First, we ignore the intensive margin
of the labor supply decision by assuming that workers are either unemployed or employed
full-time and working the same number of hours. Second, if it were not for the labor market
frictions in the matching and search-island models and the labor indivisibility in the representative family employment lotteries model, under laissez-faire everyone of working-age
would be employed. Hence, we are ignoring such non-market activities as education, child
rearing, and other types of homework that explain why some people of working-age do not
participate in the labor market.4
models are completely flexible, these wages have nonetheless been denuded of any allocating or signaling
function: this is because matching takes place before bargaining and so search effectively precedes wagesetting. . . . In conventional market situations, by contrast, firms design their wage offers in competition with
other firms to profitably attract employees; that is, wage-setting occurs prior to search so that firms’ offers
can influence workers’ search behavior and, in this way, firms’ offers can influence the allocation of resources
in the market.”
4
This means that we calibrate the representative family employment lotteries model under laissez faire
to explain an unemployment rate rather than the employment/population ratio that is more often taken as
the target by researchers using that model. See section 1.3.1 for elaboration of this point.
3
1.1
Why should macro economists care about unemployment?
Recent contributions by Prescott (2002, 2004), and Rogerson (2005a) compel us to face
a modeling issue that Lucas (1987, pp. 67-68) posed sharply when he asked “. . . whether
modeling aggregative employment in a competitive way as in the Kydland and Prescott
model (and, hence, lumping unemployment together with ‘leisure’ and all other non-work
activities) is a serious strategic error in trying to account for business cycles. I see no reason
to believe that it is. If the hours people work – choose to work – are fluctuating, it is
because they are substituting into some other activity. For some purposes, – designing an
unemployment compensation scheme, for example – it will clearly be essential to break nonwork hours into finer categories, including as one ‘activity’ unemployment. But such a finer
breakdown need not substantially alter the problem Kydland and Prescott have tried to face
of finding a parameterization of preferences over goods and hours that is consistent with
observed employment movements.”
Prescott (2002, 2004), and Rogerson (2005a) have extrapolated Lucas’s reasoning by
using a model that contains neither jobs nor a separate activity called unemployment to
explain observed differences in per capita hours in terms of differences in measures of the
pertinent flat-rate tax wedge across time and countries.5 Although UI is substantial in many
of the countries in their sample, Prescott’s and Rogerson’s calculations set UI to zero. Was
that a good idea? We think not because the big labor supply elasticities that in their models
make tax wedges so important also imply that UI will have big effects, making calculations
that ignore UI misleading.6
Prescott and Rogerson abstract from frictional unemployment. In their frictionless
economies, the idleness of some workers emerges from the combined workings of employment lotteries and complete markets for contingent claims. The random process that assigns people to work or to leisure and the complete sharing of labor income risk achieve
an economy-wide coordination of decisions absent from the other two labor market models.
Perhaps assuming such complete social cohesion is a good way to model employment fluctuations over the business cycle because, for example, U.S. companies resort to temporary
5
They also postulate that tax revenues are handed back to households either as transfers or as goods and
services, i.e., they isolate the substitution effect of taxation by arresting the wealth effect that would follow
if the revenues of those taxes were not to be rebated lump sum.
6
Hansen (1985) shows that the allocation in the employment lottery framework can be interpreted as the
outcome of a competitive equilibrium in which households can purchase arbitrary amounts of UI. Hansen’s
argument pertains to private insurance, where the representative household properly internalizes the costs
and benefits of the insurance arrangement. In contrast, when the household collects government supplied
UI, it does not internalize the costs, and this makes a big difference in outcomes. In particular, since the
representative household takes tax rates as given and independent of its own use of government supplied UI,
the household will “abuse” the UI system by choosing a socially inefficiently high probability of not working:
because the government subsidizes unemployment (leisure), households spend too much time unemployed.
As we show below, the effect is large because the labor supply elasticity is so high in the representative
family employment lotteries model. Hence, it is a mistake to think that government supplied UI facilitates
implementing an optimal allocation either by completing markets or by substituting for private insurance in
the employment lottery framework.
4
layoffs in downturns and because the experience rating in the U.S. UI system means that
the private sector at least partially facilitates consumption sharing between those who are
employed and unemployed.7 But we doubt that the abstraction of complete social cohesion
is also appropriate for understanding differences in employment observed on different sides
of the Atlantic. It is difficult for us to believe that private contractual arrangements are
what has required furloughing large numbers of European workers into extended periods of
idleness, sometimes even into absorbing states of life-time idleness. The widespread sharing
across all individuals in society that is achieved by the smooth functioning of employment
lotteries and the elaborate private transfers between households in the representative family
model contrast markedly with the friction-laded, individualistic worlds of the matching and
search-island models. We will argue that European households today are more likely to
empathize with the choices faced by workers in those latter two models, where households
must fend for themselves and seek their own fortunes in labor markets (or in government
welfare programs) while trading a limited array of financial assets that cannot replicate the
outcomes of transactions in employment lotteries and consumption claims contingent on the
outcomes of those household specific lotteries.8
To elaborate further about why macroeconomists should want to model unemployment,
consider the following two issues. First, the nature of the activities that we call unemployment and the details of the process through which jobs and workers find each other can
matter for aggregative labor market analysis. Analyzing the aggregate effects of layoff taxes
provides one good example. Lucas and Prescott (1974) concluded that a layoff tax reduces
unemployment in the search-island model, while Hopenhayn and Rogerson (1993) reached
the opposite conclusion in the representative family model with employment lotteries. We
shall investigate why the frictionless model in the latter study yields an opposite outcome
from models that embody frictional unemployment. Second, as already mentioned, designating a worker as unemployed can entitle him or her to government-provided UI that is
quite substantial in some countries. Without explicitly modelling these entitlements and
the associated labor market dynamics, we cannot properly address some puzzling observations. For example, OECD (1994, chap. 8) reports that there was a negative cross-country
correlation between UI benefit levels and unemployment in the 1960s and early 1970s. In
the early 1960s, the lower unemployment in Europe, notwithstanding UI benefits that were
more generous than in the US, attracted the attention of the Kennedy administration. The
President’s Committee to Appraise Employment and Unemployment Statistics (1962) concluded that differences in statistical methods and definitions did not explain the observed
differences and, hence, European unemployment rates were indeed lower than that of the
U.S.9 But as we know, the picture changed in the late 1970s. The deterioration of European
7
See footnote 6.
For another critical evaluation of the aggregation theory in the representative family lotteries model, see
Ljungqvist and Sargent (2004b).
9
The Committee’s verification of lower unemployment in Europe prompted Deputy Commissioner Myers
(1964, pp. 172–173) at the Bureau of Labor Statistics to write: “From 1958 to 1962, when joblessness in
[France, former West Germany, Great Britain, Italy and Sweden] was hovering around 1, 2, or 3 per cent,
[the U.S.] rate never fell below 5 per cent and averaged 6 per cent. . . . The difference between [the U.S.]
8
5
labor market outcomes is probably worse than indicated by the unemployment statistics because many unemployed Europeans have left unemployment by entering other government
programs such as disability insurance and early retirement.10
The Prescott and Rogerson versions of the representative family framework lacks frictional unemployment and government-provided UI benefits, two features that we think are
crucial for understanding the European employment experience. Instead, Prescott and
Rogerson stress employment lotteries and trading of a complete set of consumption claims
contingent on the outcomes of those lotteries, features that we find virtually impossible to
combine with realistically calibrated government-provided UI benefits in Europe: when faced
with generous government supplied UI benefits, the representative family with its high labor
supply elasticity would simply furlough far too many workers into leisure.11
1.2
An excuse for using turbulence as a laboratory
We use turbulence as our laboratory because we believe that it, along with government
supplied UI and EP, truly is an important ingredient in understanding labor market outcomes
in Europe and the U.S.12 During the 1950s and 1960s, unemployment rates were lower in
Europe than in the United States, but during the 1980s and 1990s, they became persistently
higher. In Ljungqvist and Sargent (2005), we reproduced those outcomes within models of an
artificial ‘Europe’ and an artificial ‘U.S.’ that contain identical peoples and technologies but
different labor market institutions. In Europe, as an employment protection device, there is
a tax on job destruction that is absent in the U.S., and in Europe unemployment insurance
benefits are longer and more generous than in the U.S. Our model imputes both the lower
unemployment rate and the average for these European countries was only a little more than 3 percentage
points. But, if we could wipe out that difference, it would mean 2 million more jobs, and perhaps $40 to $50
billion in Gross National Product. We can surely be excused for looking enviously at our European friends
to see how they do it. We have profited much in the past from exchange of ideas with Europe. It would be
short-sighted indeed to ignore Europe’s recent success in holding down unemployment.”
10
For example, Edling (2005) reported that in 2004 early retirees comprised 10% of the working age
population in Sweden and in 2003 2.4% had been sick for more than a year. Edling asked rhetorically
whether “unemployment is hidden in accounts other than those originally intended for the unemployed?”
The incendiary nature of his question became clear when Edling’s employer refused to publish the report,
prompting Edling to resign after having served 18 years as an investigator at the largest trade union in
Sweden.
11
The impossibility of incorporating European sized social insurance in the employment lottery framework
causes us to question the appropriateness of that abstraction for understanding the European employment
experience. Prescott (2004, p. 8) voices no such concerns but instead lauds an apparent modelling success:
“I am surprised that virtually all the large differences between the U.S. labor supply and those of Germany
and France are due to differences in tax systems. I expected institutional constraints on the operation of
labor markets and the nature of the unemployment benefit system to be of major importance.”
12
For other models of labor market institutions and turbulence, see Bertola and Ichino (1995) and Marimon
and Zilibotti (1999). Bertola and Ichino show that high EP and a rigid wage can explain why unemployment
increases in response to more volatile local demand shocks. Marimon and Zilibotti explore how high UI can
cause unemployment to rise when there is an increase in the value of forming the ‘correct matches’ between
heterogeneous workers and firms.
6
European unemployment of the 50s and 60s and the higher European unemployment of the
80s and 90s to these labor market institutions and how workers chose to respond to more
‘turbulent’ labor market prospects in the later period. We modelled turbulence by increasing
a parameter that determines an instantaneous loss of human capital by workers who suffer
an ‘involuntary’ job dissolution.13 That representation of more turbulence also generates the
increased volatilities of permanent and transitory components of earnings that have been
documented by Gottschalk and Moffitt (1994), among others.14,15
We designed our model to focus on adverse incentives and to ignore the benefits in terms
of consumption smoothing provided by social insurance: our risk-neutral workers care only
about present values of after-tax earnings and benefits.16 In the spirit of McCall (1970), our
model takes a fixed distribution of wages as exogenous: there are no firms, no bargaining
over wages, and no theory of endogenous wage determination. We added dynamics for skills
to McCall’s model and took earnings to be a wage draw times a skill level. Skills accrue
during episodes of employment and deteriorate during periods of unemployment.
Because it omits many conceivable general equilibrium effects, our extended McCall
model is fair game for criticism. How would outcomes differ in general equilibrium models that incorporate some of the features missing from our model, e.g., a matching function
that captures adverse congestion effects, firms that post vacancies, equilibrium wages that
emerge from bargaining, and a Beveridge curve; or risk-averse workers who engage in precautionary saving because of incomplete markets, firms that invest in physical capital, and a
13
Pavoni (2003) reviews a substantial body of evidence bearing witness to such human capital loss. Heckman (2003) provides a broad ranging portrayal of the European labor market institutions and outcomes that
is consistent with the idea that what we call turbulence has increased in recent decades and that that has
had adverse effects on outcomes emerging from European labor markets.
14
For a survey of the empirical evidence on increased earnings volatility, see Katz and Autor (1999).
For the United States, there is evidence that between the 1980s to the 1990s, job losses were more likely
to be associated with permanent separations rather than temporary layoffs, to cause displaced workers to
switch industries, and prompt workers to move to smaller firms which meant on average incurring permanent
losses in their wage levels. All of these indicate losses of job-specific human capital. Also, the incidence of
permanent job loss among high wage workers became larger. See Farber (1997, 2005).
15
Friedman (2005) is full of anecdotes about how reductions in communication costs and lower political
barriers, a process that can be called globalization, can lead to rapid depreciation of recently valued human
capital. For example, on page 20, Friedman quotes David Schlesinger, head of Reuters America, as saying
about New London, Connecticut: “ . . . Jobs went; jobs were created. Skills went out of use; new skills
were required. The region changed; people changed.” On page 294, Friedman remarks that workers are
paid for “general skills and specific skills” and “when you switch jobs you quickly discover which is which.”
See Jovanovic and Nyarko (1996) for an information theoretic model of human capital in which experienceacquired knowledge about parameters that calibrate a ‘task’ forms an important component of human capital
that can be destroyed when technology changes.
16
We justified the linear utility specification by referring to the very high levels of government provided
insurance in Europe. Benefits with high replacement rates and indefinite duration drastically reduce the
consumption risk faced by individual workers. Hence, our decision to abstract from risk aversion did not
seem to be too confining in the European context. The focus of our risk-neutral workers’ on the present
values of benefits can also be compared to a finding by Cole and Kocherlakota (2001) who describe a setting
in which their access to a private storage technology makes risk-averse agents evaluate alternative social
insurance contracts in terms of the present values of income that they yield.
7
competitively determined labor market? And what about the calculations by Prescott (2002,
2004) and Rogerson (2005a) that show that one can ignore the differences across countries
in UI and EP and use a model in which willing workers immediately get work with firms to
explain the important differences between labor market outcomes in Europe and the United
States?
The quantitative models in this paper answer these questions by including these general
equilibrium effects. Common features in their physical environments allow us to represent the
interactions between the transition dynamics for skills and both the unemployment benefit
levels and the job destruction taxes that Ljungqvist and Sargent (2005) focussed on, and
to assess the sensitivity of outcomes to the models’ distinct features, such as the presence
or absence of congestion effects, wage bargaining, risk aversion, and complete or incomplete
markets.
1.3
1.3.1
Calibrations
Unemployment versus an employment-population ratio
The issues about UI that preoccupy us prompt us to calibrate the labor-leisure tradeoff in
the representative family employment lotteries model to match an unemployment rate rather
than the employment to population ratio that is more often the target in the real business
cycle literature. In the laissez-faire version of the model, these different calibration targets
would have no substantial consequences other than to scale up or down the fraction of the
population of working age that is employed. However, these differences do matter in the
welfare-state versions of the model because we assume that those who are not employed are
entitled to government supplied UI. The distortionary tax rate needed to finance the UI
system is increasing in the number of people counted as unemployed. For that reason, our
decision to take the unemployment rate as the calibration target seems to be appropriate for
studying the issues about UI addressed in this paper.
1.3.2
Microfoundations
It is appropriate to issue a couple of warnings about the microeconomic foundations of the
quantitative results presented in this paper. To set out our first warning, recall how Lucas
(1987, pp. 46–47) praised Kydland and Prescott (1982) for specifying their model in terms
of parameters of preferences and technology because that enabled them to use extraneous
microeconomic evidence to calibrate its parameters. “This is the point of ‘microeconomic
foundations’ of macroeconomic models: to discover parameterizations that have interpretations in terms of specific aspects of preferences or of technology, so that the broadest range
of evidence can be brought to bear on their magnitudes and their stability under various
possible conditions. . . . here is a macroeconomic model that actually makes contact with
microeconomic studies in labor economics!”17
17
Browning et al. (1999) question calibrations of the real business cycle model for sometimes not faithfully
importing micro estimates. They also describe aggregate implications of general equilibrium theory that raise
8
However, “making contact with microeconomic studies in labor economics”, in the sense
of endowing their representative agents with the labor supply behavior of a typical individual
detected in the micro studies, was certainly not the purpose of Hansen (1985) and Rogerson
(1988) when they proposed their employment lotteries model. Instead, they showed that nonconvexities at the household level in combination with well-functioning private insurance
markets that smooth out the impact of those non-convexities on the households’ budget
constraints give rise to a high economy-wide labor supply elasticity even when the labor
supply elasticity of each of the households being aggregated is small. That makes the labor
supply elasticities that micro labor economists have estimated irrelevant for aggregative labor
market analysis.
Now for our second warning. Parts of our three models are so highly stylized that they
cannot readily be connected to micro evidence. For example, we arbitrarily specify truncated
normal distributions of productivity levels rather than calibrate them to data. However, we
can and do calibrate other parameters to match some micro observations. For example, the
earnings potential of a high skill worker is twice that of a low skill one, and it takes 10 years
on average to work your way from low skill to high skill. Note that our parameterization
for the time it takes to accumulate skills pertains both to new inexperienced workers and to
workers who have suffered skill loss and want to regain their earnings potential (see footnote
29).
1.3.3
Common parameterizations
We reiterate parameter values from previous studies but also, as far as possible, retain
common parameterizations across models. Our practice of keeping parameters fixed across
different frameworks can be criticized because the same values of these parameters imply
different outcomes in the different frameworks. This is easy to see for the discount factor,
which implies different outcomes for the risk-free interest rate in the search-island model,
where precautionary savings motives are in play, and in the representative family model,
where there are complete insurance markets.
Our justification for keeping common parameters, including the discount factor and the
variance of the productivity distribution, is that it best serves our goal of focusing attention on the economic forces at work in the alternative frameworks. And despite the single
parameterization in our study, we will argue that those economic forces are robust within a
framework either by appealing to evidence accumulated from earlier studies in the literature
or by noting that the pertinent quantitative effects are so large that no reasonable change in
parameter values can make a substantial difference. An example of the latter is our finding
that the representative family employment lotteries model delivers quantitatively unrealistic
responses to government supplied UI. This finding stems from the high aggregate labor supply elasticity that characterizes the employment lotteries model and is relatively insensitive
to the choice of parameter values.
doubts about whether the cross section and other micro empirical studies are pertinent for the parameters
of aggregative models, thereby updating arguments made by Kuh and Meyer (1957).
9
1.4
Conversations
Concerning the study of unemployment, Lucas (1987) reserved his highest praise for the
McCall search model. “Indeed, the model’s explicitness invites hard questioning.” (p. 56).
“Questioning a McCall worker is like having a conversation with an out-of-work friend:
‘Maybe you are setting your sights too high’, or ‘Why did you quit your old job before you
had a new one lined up?’ This is real social science: an attempt to model, to understand ,
human behavior by visualizing the situation people find themselves in, the options they face
and the pros and cons as they themselves see them.” We too listen to the workers and firms
in our models, especially in section 10, where we use these conversations as a way to make
up our own minds about which models ring truest in terms of the choice problems faced by
unemployed workers.18
1.5
Organization
Section 2 describes features that transcend the environments of all of our models. These
include: (1) two transition matrices for workers’ skill levels, one for workers whose jobs
continue or end voluntarily, another for workers whose jobs terminate involuntarily; (2) a
probability distribution for drawing productivity levels of new workers and transition matrices for the productivity levels of workers whose jobs continue; and (3) parameters that
define a replacement ratio for UI and a layoff tax for EP. Section 3 describes the additional
features that complete the matching models: a risk-neutral utility functional of consumption; and four sets of matching functions that define alternative market structures for sorting
workers and firms into pools where workers wait for jobs and firms post vacancies. Beyond
the common features reported in section 2, section 4 describes the additional features in the
search-island model: a discounted risk-averse functional that is separable across consumption, search effort, time, and states and that imparts a precautionary savings motive; and
firm-owned technologies for creating jobs and for converting labor and capital into output.
Section 5 describes the special features of the representative family model. Firms face the
same problem that they do in the search-island model. But there are complete markets to
smooth consumption across states and time. An infinitely lived representative family consists of a continuum of lineages composed of workers who value leisure, consumption, and
their offsprings’ welfare. Section 6 describes calibrations of the three models. Sections 7, 8,
and 9 describe calibrated outcomes in the matching, search-island, and representative family
models, respectively. Section 10 critically evaluates the mechanisms at work in the three
types of models, comments on the transcendence of the forces stressed by Ljungqvist and
Sargent (1998, 2005), and critically evaluates the representative family model and its high
labor aggregate supply elasticity as a vehicle for understanding cross-country differences in
labor market outcomes. Three appendixes, one for each model, describe Bellman equations
and equilibrium conditions.
18
Ljungqvist and Sargent (2003) reported imaginary conversations with a European and American worker
who had experienced identical sequences of labor market shocks, but who behaved differently at the ends of
their lives.
10
2
Common features of our environments
Figures 1 and 2 show the within-period timing of our models. The top halves of these
figures are identical. In all models, each of a continuum of potential workers faces a constant
probability ρ of exiting the labor force. In the matching model, a worker immediately exits
the model upon leaving the labor force while, in the other two frameworks, ρ is the probability
that a worker will retire and become unable to work, and σ is the probability that a retired
worker dies. To keep the total population and the shares of workers and retirees constant
over time, people who exit a model are replaced by newborn workers.
There are three other exogenous sources of uncertainty. First, an employed worker faces
a probability π o that his job terminates. Second, workers experience stochastic accumulation
or deterioration of skills, conditional on employment status and instances of exogenous job
terminations. Third, idiosyncratic shocks impinge on employed workers’ productivity.19
2.1
Skill dynamics
There are two possible skill levels, indexed by h ∈ {0, H}. All newborn workers enter the
labor force with the low skill index, h = 0. An employed worker with skill index h faces a
probability pn (h, h0 ) that his skill at the beginning of next period is h0 , conditional on no
exogenous job termination. In the event of an exogenous job termination, a laid off worker
with last period’s skill h faces a probability po (h, h0 ) that his skill becomes h0 . A worker’s
skill remains unchanged during an unemployment spell. The skill transition matrices are:
1 − πu πu
n
,
(1)
p =
0
1
1
0
o
.
(2)
p =
πd 1 − πd
2.2
Firm formation and productivity
The process of uniting firms and workers differs across the three frameworks but has several
common features. Firms incur a cost µ when posting a vacancy in the matching model or
when creating a job in the other two frameworks. We model a new job opportunity as a
draw of productivity z from a distribution Qoh (z). The productivity of an ongoing job is
governed by a Markov process: Qh (z, z 0 ) is the probability that next period’s productivity is
z 0 , given current productivity z. For any two productivity levels z and ẑ < z, the conditional
probability distribution Qh (z, z 0 ) first-order stochastically dominates Qh (ẑ, z 0 ), meaning that
X
X
Qh (z, z 0 ) <
Qh (ẑ, z 0 ),
for all z̄.
(3)
z 0 ≤z̄
z 0 ≤z̄
19
Endogenous job separations will impose additional risks on individual workers. Whether the workers
can fully insure against such risks varies across our models.
11
The probability distributions, Qoh (z) and Qh (z, z 0 ), depend on the worker’s skill h in the
matching model, but not in the other two frameworks.
2.3
Government mandated UI and EP
The government levies layoff taxes on job destruction and provides benefits to the unemployed. It imposes a layoff tax Ω on every endogenous job separation and on every exogenous
job termination except retirement. The government pays unemployment benefits equal to
a replacement rate η times a measure of past income. In all three models, it will suffice to
keep track of a worker’s skill in his last employment, to determine his benefit entitlement.
Newborn workers are entitled to the lowest benefit level in the economy. The government
finances unemployment benefits with revenues from the layoff tax and other model-specific
taxes.
3
Matching models
Like DenHaan et al. (2001), we include skill dynamics in a matching framework.20 The probability distributions of the productivity levels for high-skilled (h = H) workers stochastically
dominate corresponding probability distributions of low-skilled (h = 0) workers , i.e.,
X
X
X
X
QoH (z 0 ) <
Qo0 (z 0 )
and
QH (z, z 0 ) <
Q0 (z, z 0 ),
(4)
z 0 ≤z̄
z 0 ≤z̄
z 0 ≤z̄
z 0 ≤z̄
for all z̄, given that z is a permissible productivity level for both low-skilled and high-skilled
workers. We follow DenHaan et al. (2001) and assume that benefits are determined by a
replacement rate η on the average after-tax labor income in the worker’s skill category when
last employed.21 Hence, we can index a worker’s benefit entitlement by his skill in his last
employment spell, b ∈ {0, H}, so that his benefit entitlement is some function b̃(b). Let
u(h, b) be the number of unemployed workers with skill level h and skill during his previous
employment spell of b. The total number of unemployed workers is then
X
ū =
u(h, b).
(5)
h,b
20
We thank Wouter DenHaan, Christian Haefke, and Garey Ramey for generously donating their computer
code, which we have adapted. The matching framework originated in works of Diamond (1982), Mortensen
(1982), and Pissarides (2000).
21
We make two simplifications to DenHaan, Haefke, and Ramey’s specification of the benefit system. First,
newborn workers are entitled to the lowest benefit level without first having to work one period. Second,
workers who experience an upgrade in skills are immediately entitled to the higher benefit level, even if
the match breaks up immediately. These assumptions simplify solving the model. The second assumption
enables us also to discard DenHaan, Haefke and Ramey’s simplifying, but questionable, assumption that
a skill upgrade is accompanied with a new productivity draw where the lower bound on possible draws
is the reservation productivity of an ongoing match with a high-skilled worker. In our model, exogenous
distributions from which productivities are drawn do not change with endogenous reservation productivities.
12
We drop the assumption of DenHaan et al. (2001) that there is an exogenous number of
firms and instead impose a zero-profit condition that expresses the outcome of free entry.
Let v be the endogenous number of vacancies and let M (v, ū) be an increasing, concave, and
linearly homogeneous matching function:
v M (v, ū) = ū M
, 1 ≡ ū m(θ),
(6)
ū
where the ratio θ ≡ v/ū is the endogenously determined degree of “market tightness.” The
probability of finding a job, M/ū = m(θ), is an increasing function of market tightness,
and the probability of filling a vacancy, M/v = m(θ)/θ, is a decreasing function of market
tightness. We first assume a single matching function for all vacancies and all unemployed
workers, but later consider multiple matching functions.22 For various technical details, see
appendix A.
We form three models with separate matching functions (1) for unemployed workers
with different skill levels, yielding equilibrium vacancies v(h) for each h ∈ {0, H}; (2) for
unemployed workers having different benefit entitlements, yielding equilibrium vacancies v(b)
for each b ∈ {0, H}; and (3) for unemployed workers indexed by both their current skill h
and their skill b in their last employment, yielding equilibrium vacancies v(h, b) for each pair
of values (h, b) ∈ {0, H} × {0, H}. The last setup has only three matching functions because,
given our specification of the benefit system, there are no high-skilled unemployed workers
with low benefits.
We keep DenHaan, Haefke, and Ramey’s specification that workers are risk neutral.
Workers’ preferences are ordered by
E0
∞
X
β t (1 − ρ)t ct ,
(7)
t=0
where the worker discounts future utilities by the subjective discount factor β ∈ (0, 1) and
the survival probability (1 − ρ). The government finances the unemployment compensation
scheme with the revenues that it receives from the layoff tax and a flat-rate tax τ on firms’
output.
Figure 1 shows the within-period timing of events in our matching model.
3.1
Conversation with a worker and a firm
In the spirit of the conversations with McCall workers in Lucas (1987) and Ljungqvist and
Sargent (2003), it is informative to discuss the choices that confront our Diamond-MortensenPissarides worker. He likes to consume, but is indifferent to when because his utility is linear
in consumption and the inverse of the equilibrium gross interest rate equals his subjective
22
Hornstein et al. (2003) use a matching model with one matching function, workers with one skill level,
but physical capital of different vintages, as a tool for studying how different rates of embodied technical
change impinge on equilibrium outcomes, including wage distributions.
13
End of period t-1
Period t
1. Exogenous job destruction
retirement shock
exogenous job termination
2. Skill evolution
exogenously laid off
not exogenously laid off
ρ
πo
po(h, h')
pn(h, h')
3. Endogenous job formation and destruction
Firms invest in
vacancies
Unemployed wait in
matching function
Labor market, M(v, u)
old firms, Qh'(z, z')
• bilateral meetings
• firm-worker pair, Qho(z)
• Nash bargaining
So(h, z, b)
agree
Surplus functions
disagree
(no layoff tax)
S(h', z')
continue
match
lay off
(tax Ω)
Figure 1: Matching model
discount factor times his constant survival probability. When employed, he works effortlessly.
When unemployed, he waits and collects unemployment benefits without effort. When there
are more unemployed workers, he waits longer, given the number of vacancies. After being
newly matched, a firm and worker bargain about a wage. An acceptable initial bargain
commences a worker-firm relationship that lasts until one of three events occur. First, a
worker might retire. Second, nature might dissolve the match. Third, a worker and a firm
might agree to dissolve a match in response to an idiosyncratic productivity shock. In each
period that a worker and a firm remain matched, they bargain over how to split the match
surplus. UI and EP influence the surpluses for both new and continuing matches, albeit
in somewhat different ways because firms pay layoff taxes only if they had previously hired
a worker. By increasing a worker’s threat point, unemployment benefits also have a direct
impact on bargaining.
Firms choose a number of vacancies to post at a constant per-vacancy resource cost. The
matching function imposes congestion effects on both firms and workers. An increase in
vacancies decreases the waiting time between matches for each unemployed worker and increases the waiting time between matches for each vacancy, given the number of unemployed
workers. Free entry implies that firms that post vacancies can expect to earn zero profits.
Therefore, the expected discounted cost of filling a vacancy equals the expected discounted
value of a firm’s share of future match surpluses.
14
4
Search-island model
Our search-island model with incomplete markets features risk-averse workers who engage
in precautionary saving; a non trivial choice of search effort by unemployed workers; and a
competitive labor market for workers whose job searches are successful. The only vehicle
for savings is a single risk-free one-period security. Labor contracts cannot depend on the
history of a firm’s productivity. We consign various technical details to appendix B.
We create our model by altering the model of Alvarez and Veracierto (2001).23 We adopt
their specification of a worker’s preferences:
E0
∞
X
t=0
(1 − st )γ − 1
β (1 − ot ) log(ct ) + A
,
γ
t
t
with A > 0, γ > −1,
(8)
where (1 − ot ) is the survival probability with ot = ρ if the worker is of working age and
ot = σ if the worker is retired; and st ∈ [0, 1) is the worker’s choice of search intensity
if he is unemployed and of working age. A search intensity st determines an unemployed
worker’s probability sξt of finding a centralized labor market in the next period, where 0 ≤
ξ ≤ 1. Workers who find the labor market get a job paying a market-clearing wage rate. To
accommodate the feature, not present in Alvarez and Veracierto (2001), that workers differ
in their skills, we let w∗ denote the wage rate per unit of skill, where the skill level of a
low-skilled worker is normalized to one and the skill level of a high-skilled worker is 1 + H.
Hence, a low-skilled worker earns w∗ and a high-skilled worker earns (1 + H)w∗ .
4.1
Firms
We suppress Alvarez and Veracierto’s firm size dynamics and, in the spirit of our matching
model, let each firm employ only one worker. Each firm also rents physical capital. The
firm’s production function is
zt ktα (1 + ht )1−α ,
with α ∈ (0, 1),
(9)
where zt is the current productivity level, ht ∈ {0, H} is the skill index of the firm’s worker,
and kt is physical capital that depreciates at the rate δ. Output can be devoted to consumption, investment in physical capital, and startup costs. The rest of Alvarez and Veracierto’s
model of firms enters our framework as follows. Incurring a startup cost µ at time t allows a
firm to create a job opportunity at t + 1 by drawing a productivity level z from the distribution Qo (z). After seeing z, a firm decides whether to hire a worker from the centralized labor
23
The antecedent of this search-island framework is a model of Lucas and Prescott (1974) in which riskneutral workers engage in effortless but time-consuming search across a large number of spatially distinct
islands with idiosyncratic productivity shocks. The only search cost in that model is the opportunity cost of
labor income foregone when moving between islands. Alonso-Borrego et al. (2004) use a two-market version
of the Alvarez and Veracierto (2001) model to study how legal regulations that exempt fixed term labor
contracts from layoff taxes affect equilibrium outcomes.
15
market. We retain Alvarez and Veracierto’s key assumption that firms and workers first meet
under a veil of ignorance about their partner’s state vector: the firm hires a worker drawn
randomly from a single pool of unemployed workers with a mix of low-skilled and high-skilled
workers. Once hired, a firm observes a worker’s skill, hires the appropriate physical capital,
and pays the worker the market wage of w∗ per unit of skill. A firm must retain a worker
for at least one period.
Notice that the wage cannot be indexed by the history of shocks z t . Under a veil of
ignorance, all unemployed workers who have been successful in their job search efforts are
randomly matched with firms that have decided to create new jobs. The assumption that
the market-determined wage rate per unit of skill is unchanged throughout an employment
spell is restrictive. To avoid layoffs, workers would be willing to accept wage cuts in response
to some adverse productivity shocks.24
4.2
Other features
Besides markets for goods and for renting labor and capital, workers can acquire non-negative
holdings of risk-free assets that earn a net interest rate i. Following Alvarez and Veracierto
(2001), we postulate a competitive banking sector that accepts deposits that it invests in
physical capital and claims on firms. The banking sector rents physical capital to firms at
the competitive rental rate i + δ. Banks hold a diversified portfolio of all firms and so bear
no risk.
In the spirit of Alvarez and Veracierto, we assume that a worker who dies is replaced by
a newborn unemployed worker, to whom he is indifferent, but who nevertheless inherits his
assets. Newborn workers have the low skill index, h = 0.
The government pays unemployment compensation equal to a replacement rate η times
an unemployed worker’s last labor earnings net of taxes. Newborn workers are entitled to
the lowest benefit level in the economy. The government receives revenues from layoff taxes
and other revenues from an income tax whose rate we represent in terms of the difference
between before-tax and after-tax wage rates, (w∗ − w). The government balances its budget.
Figure 2 shows the within-period timing of events in our search-island model.
4.3
Conversations with a worker and a firm
Differentiated by their skill, benefit entitlements, assets, and employment-unemploymentretirement status, workers save or dissave. Employed workers supply labor effortlessly. Unemployed workers suffer more disutility when they search more intensively. After a successful
search, a worker receives a job that pays a competitively determined wage. Although the
wage rate per unit of skill is the same across all firms, a worker cares about the productivity
24
A main purpose of Alvarez and Veracierto (2001) is to quantify the potential welfare gains of a tax on
job destruction that reduces socially wasteful separations. They acknowledge that the rigidity they impose
on labor contracts and their assumption of no disutility from work cause them to overestimate those welfare
gains.
16
End of period t-1
Period t
1. Exogenous job destruction
retirement shock
exogenous job termination
2. Skill evolution
exogenously laid off
not exogenously laid off
ρ
πo
po(h, h')
pn(h, h')
3. Endogenous job formation and destruction
Firms invest in
job creation
new firms, Qo(z)
Unemployed choose
search intensities
Labor market
wage w*
Precautionary and
life-cycle savings;
risk-free asset
old firms, Q(z, z')
exit
continue
exit ≡
lay off
(tax Ω)
4. Capital rental
Capital market
(Intermediary holds
all equity and capital)
Figure 2: Search-island model
of the firm employing him. A firm’s idiosyncratic productivity shock determines when a
worker is cast into unemployment. An unemployed worker incurs the disutility from searching for a new job and consumes from his savings and whatever unemployment insurance he
is entitled to.
At a constant per-job resource cost, firms create new jobs. Free entry ensures that all
job creating firms can expect to earn zero profits. After observing the productivity of a new
job, a firm decides whether to hire a worker at the market-clearing wage rate per unit of
skill. There are no congestion effects. An unemployed worker inflicts no injury on other job
seekers beyond what a seller of a good ordinarily imposes on his competitors.
5
Representative family model with lotteries
This section describes our representative family model with lotteries, with technical details
consigned to appendix C. We make these changes to our search-island model: (1) agents care
about their offspring, so that each lineage of agents has an infinite-horizon utility function
where the survival probability is equal to one; (2) a disutility of working replaces a disutility
of search; (3) no search, or other kinds of frictions impede workers from immediately finding
the competitive labor market; and (4) agents have access to complete markets in all possible
history-contingent consumption claims. We retain the assumption of the search-island and
17
matching models that there is no ‘intensive’ margin for an agent’s choice of work, i.e., each
working-age agent faces a {0, 1} choice of working or not working an exogenously amount
of time in any given period. This indivisibility in agents’ labor supply in combination with
complete contingent claims markets can give rise to equilibrium outcomes with employment
lotteries and following Hansen (1985), and Rogerson (1988), we model the economy as a
representative family. Otherwise, from firms’ viewpoint, the labor market is identical to that
of the search-island model.
The representative family consists of a continuum of lineages indexed on the unit interval.
Each lineage consists of an infinite sequence of workers, only one of whom is alive at any
time. Each worker retires with probability ρ, then dies with probability σ. In each lineage,
a newborn worker immediately replaces a deceased worker. The representative family has a
utility function over consumption and the work of all of its lineage members:25
Z 1X
Z 1X
∞
∞
h
i
h
i
j
j
t
β log(ct ) − v(nt ) dj =
β t log(cjt ) − njt B dj ,
(10)
0
0
t=0
t=0
where cjt is lineage j’s consumption at time t and njt equals one if the current member of
lineage j is working and equals zero otherwise. We assume that v(n) increases with increases
in n. The indivisibility of the individual worker’s labor supply makes v(0) and v(1) the only
pertinent values. We let v(0) = 0 and v(1) = B, so the parameter B > 0 captures the
disutility of working.
The technology and choices of a typical firm are identical with those in the search-island
model described in section 4.1. So while there are no search or matching frictions for workers,
a friction in the form of the job-creation cost µ still confronts firms. In each period, the arrival
order of shocks is 1) exogenous job destruction due to retirement shock with probability ρ
and exogenous job termination with probability π o , and 2) skill evolution. Then firms and
families take actions, and job seekers are matched with vacancies immediately and without
friction. A firm hires a worker under a veil of ignorance about his skill and rents physical
capital after seeing the worker’s skill.
5.1
Conversations with a worker and a firm
Each family member does what he is told.26 A collective entity called ‘the family’ determines
fractions Rt , U0t , U4t , UHt , N0t , NHt of workers who are retired, unemployed with low skills
and low benefits, unemployed with low skills and high benefits, unemployed with high skills
and high benefits, employed with low skills, and employed with high skills, respectively;
Rt + U0t + U4t + UHt + N0t + NHt = 1. The family adjusts these fractions in response to wage
25
It is rewarding to compare the representative family model of Hansen (1985) and Rogerson (1988) with
the household labor supply model of Chiappori (1992, 1997), and Browning and Chiappori (1998), and how
differently they draw the line that separates families. In the representative family model, a family is an
aggregate (national) economy. In applications of the household labor supply model, a family consists of
what we usually think of as a nuclear family.
26
Alternatively, the family member responds to equilibrium prices. See Hansen (1985).
18
rates, unemployment benefits and taxes. Each worker participates in an employment lottery
that determines whether he works as his personal history of skill and benefit entitlement
unfolds. Winners of the lottery do not work. As in the models of Hansen (1985) and Rogerson
(1988), the employment lotteries enlarge the aggregate labor supply elasticity relative to what
would be chosen by an individual worker with preferences under the integral sign on the left
side of (10) who did not face the {0, 1} restriction on his choice of njt .27
Because (10) is additively separable, the family assigns the same consumption c̄t to every
family member. The family assigns an unretired person to work with a probability that
depends on his or her skill and benefit entitlement.28 The unconditional expected utility of
a lineage in the representative family is
∞
X
t=0
h
i
β t log(c̄t ) − n̄t B ,
(11)
where n̄t is the unconditional probability that a member of the lineage works in period
t, which equals the fraction of all family members of both skill levels sent to work, n̄t =
N0t + NHt . Reformulation (11) of the family’s utility function (10) embeds the outcomes
of both the work-sharing lottery and trades in a comprehensive set of markets in history
contingent claims to consumption.
The technologies and choices confronting firms are the same as in the search-island model.
At a cost µ, a new one-job firm can be formed. After observing the productivity of new jobs,
firms can choose to hire workers under a veil of ignorance and pay a market-clearing wage
rate per unit of skill. Upon a non-retirement separation from a worker, a firm pays the layoff
tax.
6
Calibrations
An employed worker keeps his productivity from last period with probability (1 − π) and
draws a new productivity with probability π from the distribution Qoh (z 0 ), so that new
productivities on existing jobs are drawn from the same distribution as the productivities at
the time of job creation; Qoh (z 0 ) depends on the worker’s current skill index h in the matching
model, but not in the other two frameworks.
27
The aggregate labor supply elasticity will be altered if we expand the number of points in the set of hours
that an individual worker is allowed to work, thereby somewhat relaxing the severe indivisibility imposed in
the original models of Hansen (1985) and Rogerson (1988). See the extension to handle straight-time and
overtime work by Hansen and Sargent (1988).
28
Relative to Hansen (1985) and Rogerson (1988), we extend the objects that the employment lottery
randomizes across to include entire histories (actually, it is better to call them ‘futures’) of skills and benefit
entitlements that a worker might experience. For example, a worker who experiences a skill upgrade will
never again be assigned to enjoy leisure, at least as long as he does not suffer an involuntary layoff with skill
loss. In the spirit of Lucas (1987, p. 67, footnote 13) when he remarked that “in the Hansen and Rogerson
papers, [unemployed] workers are happier than those who draw employment!,” we can say that the real losers
of our lotteries are the workers who have had successful careers with skill accumulation because they enjoy
the same consumption as everyone else but none of the leisure. (For details, see appendix C.7.)
19
We turn first to a set of parameter values that are common to all models. Thereafter,
we report calibrations of features that are unique to each framework. As far as possible,
we reiterate parameter values from previous studies. For calibrating labor market frictions
and disutilities of searching and working, we target a laissez-faire unemployment rate in the
range of 4 to 5%.
6.1
Parameter values common to all models
Following Alvarez and Veracierto (2001), we set the model period equal to half a quarter,
and specify a discount factor β = 0.99425 and a probability of retiring ρ = 0.0031 that
are the same in all three frameworks. People of working age have an annualized subjective
discount rate of 4.7%. On average, they spend 40 years in the labor force.
Table 1 shows that the skill accumulation process is the same across models. We set
transition probabilities to make the average durations of skill acquisition and skill deterioration agree with those in Ljungqvist and Sargent (1998, 2005), who let it take a long time to
acquire the highest skill level in order to match realistic shapes of wage-experience profiles.29
We set a semiquarterly probability of upgrading skills π u = 0.0125, so that it takes on average 10 years to move from low to high skill, conditional on no job loss. Exogenous layoffs
occur with probability π o = 0.005, i.e., on average once every 25 years. The probability of a
productivity switch on the job equals π = 0.05, so that a worker expects to retain a given
productivity level for 2.5 years.
Another common assumption is that productivities are drawn from a truncated normal
distribution with mean 1.0 and standard deviation 1.0. Model-specific assumptions dictate
how these productivity draws enter the production technology.
6.2
Matching models
Here we adopt most of the parameter values of Ljungqvist and Sargent (2004a), who modify
the matching framework of DenHaan et al. (2001). The calibration is reported in Table 1.
The main substantial departures from Ljungqvist and Sargent (2004a) are that (1) we replace
the earlier uniform productivity distributions by truncated normal distributions; (2) instead
of a fixed number of firms, we assume free entry; and (3) we introduce a Cobb-Douglas
matching function and a vacancy cost µ.30
High-skilled workers’ productivity distribution is a truncated N (2, 1) and low-skilled
workers’ productivity distribution is a truncated N (1, 1), both of which are rescaled to inte29
We thank Dan Hamermesh for conversations about his data explorations of wage-experience profiles.
Our assumption that work experience alone can double a worker’s earnings seems to line up well with data
for full-time male workers in the U.S. manufacturing industry. But the time required to attain such earnings
gains are longer than what we assume. Note that the speed of skill accumulation in our model pertains to
both new inexperienced workers and workers who have suffered skill loss and want to regain their earnings
potential.
30
To ensure that the Cobb-Douglas matching technology generates permissible matching probabilities
inside the unit interval, we assume that the number of matches equals min{M (v, ū), v, ū}.
20
Parameters common to all models
Discount factor β
Retirement probability ρ
Probability of upgrading skills, π u
Probability of exogenous breakup, π o
Probability of productivity change, π
Productivity distribution
0.99425
0.0031
0.0125
0.005
0.05
truncated N (1, 1)
Additional parameters in matching models
Matching function, M (v, u)
0.45 v 0.5 u0.5
Vacancy cost, µ
0.5
Worker’s bargaining weight, ψ
0.5
Low-skilled workers’ productivity:
truncated N (1, 1)
High-skilled workers’ productivity:
truncated N (2, 1)
Parameters common to search-island and representative family model
Probability of dying, σ
0.0083
Capital share parameter, α
0.333
Depreciation rate, δ
0.011
Job creation cost, µ
5.0
Low skill level
1.0
High skill level, (1 + H)
2.0
Additional parameters in search-island model
Disutility of search, A
5.0
γ
0.98
Search technology, ξ
0.98
Additional parameter in representative family model
Disutility of working, B
1.01
Table 1: Parameter values (one period is half a quarter)
21
grate to one. Both distributions are truncated to a range of 4 units where the midpoint of
the range is the mean of the corresponding untruncated distributions. Thus, the range for
high-skilled workers’ productivities is [0,4] and the range for low-skilled workers’ productivities is [-1,3]. The high-skilled workers’ distribution is the low-skilled workers’ distribution
shifted to the right.
Table 1 shows that our parameterization of the matching technology and the Nash bargaining between workers and firms is fairly standard. Workers’ bargaining weight equals
ψ = 0.5, which also equals the matching elasticity of the Cobb-Douglas matching function.
By computing the expected cost θµ/m(θ) of filling a vacancy, we can interpret the semiquarterly vacancy cost µ = 0.5. In the laissez-faire economy, this average recruitment cost
equals 3.4, which can be compared to the average semiquarterly output of 2.3 goods per all
workers. Our calibration of the matching model yields a laissez-faire unemployment rate of
5.0%.
6.3
Search-island model
In addition to the discount factor and the probability of retiring, we take several other
parameter values from Alvarez and Veracierto (2001): see our Table 1. We take the following
survival, technology, and preference parameters from them: {σ, δ, ξ, γ}. Since the model
period equals half a quarter and the survival probability in retirement equals σ = 0.0083, the
average duration of retirement is 15 years. The semiquarterly depreciation rate is δ = 0.011.
Our settings of exponents on the search technology (ξ = 0.98) and on the disutility of search
(γ = 0.98), respectively, make these close to linear.
One-worker firms operate a constant-returns-to-scale Cobb-Douglas production technology with a capital share parameter α = 0.333. Each firm has an idiosyncratic multiplicative
productivity shock that is drawn from a distribution that is generated by truncating N (1, 1)
to the interval [0,2] and then rescaling it to integrate to one. Low-skilled workers have one
unit of human capital while high-skilled workers have twice that amount, (1 + H) = 2.
The cost of starting a firm, i.e., of drawing anew from the distribution of productivities,
equals 5. This can be measured against the laissez-faire outcome that only about 20% of all
such draws exceed the optimally chosen reservation productivity when the firm hires a worker
at a semiquarterly equilibrium wage rate equal to 6.4 for low-skilled workers. Hence, the
average cost of recruiting a worker is approximately 6 months of the wage paid a low-skilled
worker.
The disutility parameter A for job search equals 5, which generates a laissez-faire unemployment rate of 4.4%.
6.4
Representative family model with lotteries
Table 1 shows that the representative family model and the search-island model are calibrated
alike, except for parameters pertaining to job search. There is neither a search technology
22
nor a disutility of searching. Instead, the new parameter B represents disutility of working.
Setting B = 1.01 makes the laissez-faire unemployment rate be 4.7%.
7
Quantitative findings in matching models
This section reports the effects of increases in turbulence on equilibrium outcomes in four
matching models that differ in how they sort workers before assigning them to a matching
function. How outcomes in these four models respond to increased turbulence illuminates the
economic forces that equilibrate labor markets within matching models. Matching models
feature adverse congestion effects that job-seeking workers impose on each other and that
worker-seeking firms impose on each other. Unmatched workers and firms are concerned
about both matching probabilities that are affected by the total stocks of unemployment
and vacancies, and the bargaining situation that they will face in future matches. Within
a labor pool defined by a matching function, market tightness, v/u ≡ θ, is an important
equilibrating variable that the invisible hand uses to reconcile the decisions of firms and
workers.
7.1
Single matching function
In the model with a single matching function, figure 3 shows that the unemployment rate
is positively related to the replacement rate in the unemployment insurance system. This
result emerges in almost any model of unemployment, but it is useful to recall the forces
that produce this outcome in the matching model. Unemployment benefits raise the value
of a workers’ outside option in the wage bargaining with employers. If nothing else changed,
a higher threat point for workers would cause wages and the reservation productivity to
rise. That would deteriorate firms’ bargaining positions, leaving them unable to recover the
expected cost of filling vacancies if their probability of encountering unemployed workers
were to remain unchanged. Therefore, the invisible hand restores the profitability of firms
by lowering the number of vacancies relative to the number of unemployed workers, i.e.,
the equilibrium measure of market tightness falls, which in turns implies a longer average
duration of unemployment spells. Hence, unemployment rises because the duration and
incidence of unemployment both increase.
Although UI benefits necessarily increase unemployment in the matching model, layoff
taxes have countervailing effects on unemployment. Mortensen and Pissarides (1999) pointed
out that layoff taxes reduce incentives both to create jobs and to destroy them. They show
that the net effect of these forces on market tightness, and consequently on unemployment
duration, is ambiguous, but that the reservation productivity for existing jobs decreases and
therefore so does the incidence of unemployment.31 Figure 4 shows that in our calibrated
31
The ambiguous effect of layoff taxes upon unemployment is compounded further in our framework
because we model a new job opportunity as a draw from a productivity distribution, so that there is an
endogenous reservation productivity in job creation. In contrast, Mortensen and Pissarides (1999) assume
that all new jobs begin with the same productivity level.
23
12
Unemployment rate (per cent)
10
8
6
4
2
0
0
0.1
0.2
0.3
0.4
Replacement rate η
0.5
0.6
0.7
Figure 3: (Matching model) Unemployment rates for different replacement rates η, given
tranquil economic times and no layoff taxes.
matching model with one matching function there is a strong negative relationship between
layoff taxes and unemployment. This is also true for the calibration of Mortensen and
Pissarides (1999). Unemployment falls because layoff taxes reduce labor reallocation and
lock workers into their jobs, so that frictional unemployment falls. Although this negative
relationship between layoff taxes and unemployment is the predominant outcome in the
matching literature, there are exceptions, most notably Millard and Mortensen (1997). As
explained by Ljungqvist (2002), such contradictory quantitative findings in the matching
literature come not from differences in parameter values but from different assumptions
about bargaining strengths. Millard and Mortensen (1997) assume that firms must also
pay layoff taxes after encounters with job seekers who are not hired. That dramatically
increases workers’ bargaining strengths, making equilibrium market tightness plummet in
order to level the playing field for firms. Under the more typical assumption that firms pay
layoff taxes only for the workers they had chosen to hire and then had subsequently laid
off, Ljungqvist (2002) concludes on the basis of a wide range of simulations that there is a
presumption that layoff taxes reduce unemployment in the matching model.32
7.1.1
High layoff taxes and high benefits
As noted by Mortensen and Pissarides (1999), the countervailing forces of unemployment
benefits and layoff taxes in the matching model can explain why the unemployment rate in
a welfare state need not be high. For low values of the turbulence parameter π d , a more
32
Unlike Millard and Mortensen (1997), our models obey the dictum, “If a firm does not hire, it does not
have to fire.”
24
5.5
5
Unemployment rate (per cent)
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
0
5
10
Layoff tax Ω
15
20
25
Figure 4: (Matching model) Unemployment rates for different layoff taxes Ω, given tranquil
economic times and no benefits. The magnitude of the layoff tax can be compared to an
average semiquarterly output of 2.3 goods per worker in the laissez-faire economy, i.e., a
layoff tax equal to 19 corresponds to approximately one year’s of a worker’s output.
generous unemployment insurance system can accompany higher layoff taxes, leaving the
equilibrium unemployment rate unchanged or even lower than the laissez-faire outcome. To
illustrate this outcome, we pick a replacement rate of 70%, which on its own would have
raised the unemployment rate from the laissez-faire level of 5.0% to 12.3% in figure 3, and we
choose a layoff tax equal to 24, which corresponds to approximately 5 quarters of a worker’s
average output in laissez faire, which on its own would have lowered the unemployment rate
from 5.0% to 1.9% in figure 4. The combination of these two policies yields an unemployment
rate of 4.4%, which falls below the laissez-faire unemployment rate of 5.0%, as depicted in
figure 5 at zero turbulence, π d = 0. This is qualitatively the same outcome as in the analysis
of the European unemployment experience in the 1950s and 1960s in Ljungqvist and Sargent
(2005).
7.1.2
Turbulence and unemployment
In addition, the matching model confirms the finding of Ljungqvist and Sargent (1998, 2005)
that increased turbulence causes unemployment to increase in the welfare state while it remains virtually unchanged in the laissez-faire economy, as shown in figure 5. The figure
attributes the unemployment increase in the welfare state to the generous UI system, because government mandated EP on its own would not have caused unemployment to rise.
Furthermore, the positive relationship between turbulence and unemployment is explained
by the choices made by laid off workers who have suffered skill loss, as detailed in the left
25
16
Unemployment rate for (η, Ω)
14
(.7, 0)
12
10
(.7, 24)
8
(0, 0)
6
4
(0, 24)
2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Turbulence
0.7
0.8
0.9
1
Figure 5: (Matching model) Unemployment rates for different government policies as indexed
by (η, Ω) where η is the replacement rate and Ω is the layoff tax. The solid line indexed by
(0, 0) refers to the laissez-faire economy.
panel of figure 6.
The left panel of figure 7 shows that in the welfare state an increase in turbulence increases
the average duration of unemployment spells but leaves the inflow rate almost unchanged.
The higher average duration of unemployment is not shared equally among unemployed
workers. Although all unemployed workers face the same probability of encountering a
vacancy because they enter a common matching function, job acceptance rates differ among
workers who are heterogenous with respect to their skill levels and benefit entitlements.
Thus, consider unemployed workers who have been laid off and suffered skill loss. Because
unemployment benefits are indexed to past earnings, such workers receive benefits that are
high compared to their current earnings potential. To give up their generous benefits, these
workers must encounter vacancies with idiosyncratic productivities that are high enough to
induce firms to offer more generous wages. Hence, low-skilled unemployed workers with high
benefits encounter fewer acceptable matches than do low-skilled unemployed workers with
low benefits. The unchanging inflow rate into unemployment is explained by the nearly
constant reservation productivities that determine job destruction.Turning to the laissezfaire economy in the right panel of figure 7, both the inflow rate and the average duration of
unemployment are virtually unaffected by turbulence. In the laissez-faire economy, firms and
workers respond to turbulence in ways that leave both the optimal rate of job destruction
and the optimal length of time to search for a job unchanged.
Since turbulence sharply increases the average duration of unemployment spells in the
welfare state, after allowing for the equilibrium response in the reservation productivity for
new jobs, one would expect a precipitous fall in market tightness θ = uv . Thus, the dotted
26
10
9
9
8
8
Unemployment rate (per cent)
Unemployment rate (per cent)
10
7
6
5
4
3
7
6
5
4
3
2
2
1
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Turbulence
0.7
0.8
0.9
0
0
1
0.1
(a) Welfare state
0.2
0.3
0.4
0.5
0.6
Turbulence
0.7
0.8
0.9
1
(b) Laissez-faire economy
5.5
5.5
5
5
Inflow rate (per cent) and duration (quarters)
Inflow rate (per cent) and duration (quarters)
Figure 6: (Matching model) Unemployment rates in the welfare state (panel a) and the
laissez-faire economy (panel b). The solid line is total unemployment. The dashed line
shows the unemployed who have suffered skill loss. The policy of the welfare state is (η, Ω) =
(0.7, 24).
4.5
4
Duration
3.5
3
2.5
Inflow rate
2
1.5
1
0.5
0
0
4.5
4
3.5
3
Inflow rate
2.5
2
1.5
Duration
1
0.5
0.1
0.2
0.3
0.4
0.5
0.6
Turbulence
0.7
0.8
0.9
1
(a) Welfare state
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Turbulence
0.7
0.8
0.9
1
(b) Laissez-faire economy
Figure 7: (Matching model) Inflow rate and average duration of unemployment in the welfare
state (panel a) and the laissez-faire economy (panel b). The dashed line is the average
duration of unemployment in quarters. The solid line depicts the quarterly inflow rate into
unemployment as a per cent of the labor force. The policy of the welfare state is given by
(η, Ω) = (0.7, 24).
27
1.6
1.4
Low skills (current)
Market tightness
1.2
1
#1
0.8
0.6
High skills (current)
0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Turbulence
0.7
0.8
0.9
1
Figure 8: (Matching model) Market tightness θ when there are separate matching functions
for unemployed based upon their current skills; low skills (solid line) and high skills (dashed
line). As a benchmark, the dotted line labelled #1 depicts market tightness in the economy
with a single matching function. The government’s policy is given by (η, Ω) = (0.7, 24).
line in figure 8 depicts how market tightness plummets in response to higher turbulence. To
ensure that firms break even when posting vacancies in a more turbulent environment, the
invisible hand increases the probability that a vacancy encounters an unemployed worker,
and thereby weakens the effective bargaining strength of workers: the lower probability that
an unemployed worker encounters a vacancy causes workers’ outside value to fall. Decreased
waiting times between matches for vacancies and the associated fall in a worker’s outside
value are how the invisible hand improves firms’ prospects in response to two adverse forces on
firms’ profits. First and foremost, increased turbulence ignites adverse welfare-state dynamics
because with a given replacement rate UI becomes more valuable compared to what can be
earned working. This is most apparent in the case of laid off workers who have suffered
skill loss and become low-skilled unemployed workers who are entitled to benefits that are
generous relative to their reduced earnings potential. The invisible hand must compensate
firms for meeting such workers, because these encounters are less likely to result in agreeable
matches; and when such matches are formed, wage payments to low-skilled workers who are
entitled to high benefits are higher than those to low-skilled workers who are entitled to
low benefits. Second, our representation of turbulence implies a worse technology for skill
accumulation and, therefore, higher turbulence has detrimental effects on match surpluses
in both the welfare state and the laissez-faire economies. The invisible hand must improve
firms’ situations because they have to break even while financing the average cost for filling
a vacancy out of a fixed fraction of the diminished match surpluses.
Of these two forces that make market tightness fall in response to increased turbulence,
28
that driven by adverse welfare-state dynamics is more important. This assertion emerges
from the outcome that laissez-faire unemployment in the right panel of figure 6 increases
only slightly in response to increased turbulence: under laissez-faire, the second adverse
force from increased turbulence operates, but not the first.
7.2
7.2.1
Multiple matching functions
Separate matching functions for different skills
Figure 8 also reports outcomes of an economy with two separate matching functions for
unemployed workers sorted only according to their current skills. It is instructive to compare
the market tightness across the two labor markets in such an economy. When π d = 0,
the zero-profit condition for job creation calls for more vacancies per unemployed worker
assigned to the low-skill market. Workers with low skills enjoy a higher probability of
encountering a vacancy because there is a larger match surplus to be shared in the case
of a match and therefore more incentive for firms to post vacancies. The larger match
surplus associated with a low-skilled worker arises from the possibility that employment
might result in a skill upgrade that leads to a future capital gain for the match. However,
the relative advantage for low-skilled unemployed in terms of market tightness erodes quickly
as turbulence increases. When there is turbulence, the low-skill market includes not only
low-skilled workers with low benefits but also laid off workers who have suffered skill loss
and are now low-skilled but entitled to high benefits. As discussed above, firms think that
low-skilled workers who are entitled to high benefits are poor job candidates and, therefore,
the invisible hand must compensate firms that post vacancies in the low-skill market by
assigning shorter times to encounter an unemployed worker. Thus, there has to be lower
market tightness and an associated weakening of workers’ effective bargaining strength. In
terms of aggregate unemployment, figure 11 shows that the outcome in this economy with
two matching functions is not much different from the model with a single matching function.
7.2.2
Separate matching functions for different unemployment benefits
In figure 9, we turn to an economy with separate matching functions for unemployed workers
sorted only according to their benefits. The least desirable job candidates, the low-skilled
workers entitled to high benefits, are now pooled with the high-skilled unemployed. When
turbulence π d increases, the high-benefit market experiences a precipitous fall in market
tightness, while the decline in the low-benefit market is smaller, at least until turbulence
reaches a critical level. When turbulence reaches 0.60, marked by a star in Figure 9, the
market tightness θ = uv in the high-benefit market has fallen so much that the probability
that a vacancy meets an unemployed worker is equal to one. Higher levels of turbulence
further depress market tightness and reduce the probability that an unemployed worker
meets a vacancy, but the probability that a vacancy meets an unemployed worker remains
one. The short end of the market, the number of vacancies, determines the total number
of encounters. When turbulence reaches another critical level at 0.96 marked by a circle
29
1.6
Low benefits
1.4
Market tightness
1.2
1
#1
0.8
0.6
0.4
High benefits
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Turbulence
0.7
0.8
0.9
1
Figure 9: (Matching model) Market tightness θ when there are separate matching functions
for unemployed workers sorted according to their benefits; low benefits (solid line) and high
benefits (dashed line). As a benchmark, the dotted line labelled #1 depicts market tightness
in the economy with a single matching function. The government’s policy is (η, Ω) = (0.7, 24).
For π d above .6 denoted by the ∗ , the probability that a vacancy meets a worker equals 1.
For π d above .96 (the circle), the high benefit market closes.
in figure 9, market tightness in the high-benefit market has fallen to zero and the market
closes. At levels of turbulence above this critical point, a firm in the high-benefit market
cannot expect to break even by posting a vacancy and meeting a worker with certainty in the
next period, even though the worker’s threat point is merely the outside value of remaining
unemployed forever. Vacancies in the high-benefit market have become unprofitable because
the odds of encountering a low-skilled worker rather than a high-skilled worker are too high.
The point at which the high-benefit market breaks down obviously depends on the length
of a model period. While a shortening of our semiquarterly period would delay and maybe
eliminate the market breakdown, unemployment would still explode when higher turbulence
drives market tightness closer to zero, as further discussed in section 7.2.5.
7.2.3
Comparisons across different pooling arrangements
Figure 11 depicts how aggregate unemployment increases more in response to turbulence
in the economy with separate matching functions indexed by unemployed workers’ benefits
than it does in the economy with a single matching function or in the economy with separate matching functions for unemployed workers sorted according to their current skills.
In all three economies, low-skilled unemployed workers entitled to high benefits harm the
prospects of other workers with whom they are pooled within a matching function. However,
30
the impact of these unwanted job candidates is diluted when there is a single matching function in the economy or, in the case of multiple matching functions, when these workers are
pooled with a group of unemployed workers who can better withstand such a mixing. The
low-skilled workers entitled to low benefits are the more resilient group of unemployed because their match surpluses include the prospects of capital gains associated with becoming
high-skilled: if it is worthwhile to assign any workers to operate the economy’s technology,
it must show up in the match surplus of newborn workers. Thus, low-skilled unemployed
workers entitled to low benefits are the most resilient group of unemployed in terms of bearing the burden of being pooled with low-skilled unemployed entitled to high benefits. If
these unwanted job candidates are instead pooled with the less resilient group, namely, the
high-skilled unemployed workers who are entitled to high benefits, as when the matching
functions are indexed by benefits b but not skills h, aggregate unemployment increases more
with turbulence and increases further at the level of turbulence where the probability of that
a vacancy meets an unemployed worker in the high-benefit market has increased to its maximum of one, indicated by a star in figure 11. At higher levels of turbulence, the ever lower
measures of market tightness in figure 9 cause aggregate unemployment virtually to explode
in figure 11.33 At the critical level of turbulence indicated by a circle, the high-benefit market
shuts down and unemployment becomes an absorbing state for all skilled workers who suffer
exogenous layoffs. It follows that the solutions to workers’ and firms’ optimization problems
are no longer affected by the incidence of skill loss among the exogenously laid off. Hence,
unemployment in figure 11 and measures of market tightness in figure 9 become constant
for any turbulence above this critical level. A consequence of the adverse outcomes in the
high-benefit market is that market tightness in the low-benefit market suffers a dramatic
decline. The breakdown of the high-benefit market is tantamount to a drastic deterioration
in the skill accumulation technology. The argument in section 7.1.2 explains why the invisible hand must lower market tightness in the low-benefit market to uphold the zero-profit
condition for job creation in the face of what is analogous to a deterioration in the economy’s
technology.
7.2.4
Three matching functions: sorting according to both skills and benefits
Insights gleaned from the models with two matching functions help to understand the outcomes for the model in which three matching functions sort unemployed workers perfectly
along all of their attributes. Under our calibration, the labor market for low-skilled workers
who are entitled to high benefits operates only at very low levels of turbulence, and its market tightness in figure 10 is so low that the probability that a vacancy meets an unemployed
worker always equals one. At the critical level of π d = 0.09 indicated by a circle in figure
10, the market for low-skilled unemployed workers entitled to high benefits shuts down, just
33
The unmarked point of inflection at turbulence 0.66 on aggregate unemployment in the economy with
separate matching functions for unemployed workers based upon their benefits in figure 11 coincides with the
endogenous job destruction involving high-skilled workers coming to an end. At higher levels of turbulence,
high-skilled workers separate from their jobs only because of exogenous job destruction. This equilibrium
outcome somewhat arrests the explosion in unemployment.
31
1.6
1.4
1.2
Market tightness
Low skills/benefits
1
#1
0.8
0.6
0.4
High skills/benefits
0.2
Low skills and high benefits
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Turbulence
0.7
0.8
0.9
1
Figure 10: (Matching model) Market tightness θ when there are separate matching functions
for unemployed based upon both their current skills and benefits; low skills/benefits (upper
solid line), high skills/benefits (dashed line) and low skills but high benefits (lower solid
line). As a benchmark, the dotted line depicts market tightness in the economy with a single
matching function. The government’s policy is (η, Ω) = (0.7, 24).
as the high-benefit market did in the economy of subsection 7.2.2. The two active labor
markets are characterized by smoothly falling measures of market tightness throughout the
range of turbulence in figure 10. This particular outcome resembles the outcome of the
single matching function model of section 7.1, but here the similarities end. The unemployment rate of the model with three matching functions explodes in response to increased
turbulence in figure 11, both before and after the breakdown of the market for low-skilled
unemployed entitled to high benefits. This is hardly surprising after the market breakdown
because unemployment has then become an absorbing state for all workers experiencing skill
loss. Hence, the ranks of the unemployed must then inevitably swell in response to higher
incidence of skill loss.
7.2.5
A semimonthly model period
To illustrate how the breakdown of markets depends on the length of a model period, we
temporarily adopt the assumption of a semimonthly rather than a semiquarterly model
period.34 The shorter model period makes the invisible hand less disposed to shut down
34
We convert our semiquarterly calibration into a semimonthly one by rescaling the discount factor, all
exogenous transition probabilities, and the multiplicative coefficient in the matching function. After doing so,
the laissez-faire outcome is virtually unchanged, with the qualification that the similar numerical outcomes
for output and wages now refer to a semimonthly frequency. Hence, to keep any layoff tax in the welfare
32
markets. The condition for a market breakdown now becomes that it is unprofitable to post
a vacancy even if an unemployed worker is encountered with certainty after merely half a
month rather than half a quarter as under our semiquarterly calibration.
Figure 12 depicts aggregate unemployment outcomes for both the semimonthly and the
semiquarterly calibrations of the two models that exhibited market breakdowns under the
semiquarterly calibrations. Starting with the economy with separate matching functions for
unemployed workers sorted according to their benefits, we see how the shorter model period
delays the market breakdown until a higher level of turbulence. Given the semimonthly calibration, the invisible hand finds it relatively easier to compensate firms that post vacancies
in the market for unemployed workers entitled to high benefits by pulling the measure of
market tightness ever closer to zero. This manifests itself by making unemployment rise at an
ever increasing rate until the market finally shuts down. In the economy with three matching
functions, the shorter model period results in an interior outcome for the probability that a
vacancy meets an unemployed worker in the market for low-skilled workers who are entitled
to high benefits until turbulence reaches the critical level of 0.23 that is indicated by a star
in figure 12. At higher levels of turbulence, the invisible hand can improve firms’ situation
in that market only by lowering the probability that an unemployed worker meets a vacancy
while keeping the probability that a vacancy meets an unemployed worker equal to one. Under those circumstances, we see that the market quickly shuts down at the point indicated
by a circle in figure 12, without any perceptible range of turbulence between the star and
the circle. (This contrasts with the outcome in the economy with two matching functions
where the unemployed workers were sorted according to their benefits.) At this point, the
aggregate unemployment rate immediately reaches the trajectory of our earlier semiquarterly calibration. Recall that the market for low-skilled workers entitled to high benefits in
effect isolates the workers whom firms regard as poor job candidates. Not surprisingly, the
invisible hand sets a low market tightness in this market, implying a high probability that a
vacancy encounters an unemployed worker and a low probability that an unemployed worker
encounters a vacancy. This market is truly characterized by long-term unemployment and
its average duration of unemployment spells spirals ever higher as turbulence increases, until
the market eventually breaks down.
7.2.6
Heterogeneity not duration dependence as source of falling hazard rate
For the semiquarterly calibration, figure 13 depicts the hazard rate of gaining employment
in the most turbulent times (π d = 1.0). The hazard rate is practically flat in the laissezfaire economy but declines sharply in the welfare state. The high incidence of long-term
unemployment in the welfare state is conveyed graphically by a hazard rate that is low
even at the start of unemployment spells. Compare this to the much higher and constant
hazard rate in the laissez-faire economy. The hazard rate in the welfare state falls with the
duration of spells entirely because of heterogeneity, not duration dependence. Our model
state comparable across calibrations, the tax in the semimonthly calibration has to be set thrice as large as
the tax in the semiquarterly calibration.
33
0.5
0.45
0.4
0.35
#3
0.3
# 2 (benefits)
0.25
Unemployment rate for given # of matching functions
0.2
0.16
0
0.14
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.12
# 2 (current
skills)
0.1
#1
#3
0.08
# 2 (benefits)
0.06
0.04
0.02
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Turbulence
0.7
0.8
0.9
1
Figure 11: (Matching model) Aggregate unemployment rates for different number of matching functions. The lower solid line depicts the benchmark model with one matching function.
The dash-dotted and the upper solid lines refer to the two models with two matching functions where the unemployed are sorted by their current skills and their benefits, respectively.
The dashed line depicts the model with three matching functions, i.e., the unemployed are
perfectly sorted along all of their attributes. The government’s policy is (η, Ω) = (0.7, 24).
34
0.5
0.45
0.4
0.35
0.3
# 2 (benefits)
semiquarterly
0.25
#3
semiquarterly
Unemployment rate for given # of matching functions
0
0.14
semi−
monthly
semimonthly
0.2
0.16
0.1
0.2
0.3
0.4
#3
semimonthly
0.5
0.6
0.7
0.8
0.9
1
0.12
0.1
#3
semi−
quarterly
# 2 (benefits)
semiquarterly
# 2 (benefits)
semimonthly
0.08
0.06
0.04
0.02
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Turbulence
0.7
0.8
0.9
1
Figure 12: (Matching model) How aggregate unemployment rates are affected in the welfare
state by the length of a period in the model with two matching functions where the unemployed are sorted by their benefits, and in the model with three matching functions. The
dashed lines refer to the semiquarterly calibration of figure 11, and the solid lines depict the
semimonthly version of that calibration.
35
0.5
Laissez faire
#1
#3
Hazard rate of gaining employment (semiquartely)
0.45
0.4
0.35
0.3
0.25
0.2
0.15
Welfare state
0.1
#1
0.05
0
0
#3
2
4
6
Time (quarters)
8
10
12
Figure 13: (Matching model) Semiquarterly hazard rates of gaining employment in turbulent
economic times, π d = 1.0. The two dashed lines indexed by #1 and #3 depict the welfare
state with one and three matching functions, respectively. The solid line represents the
laissez-faire economy with an almost perfect overlap of the outcomes associated with one and
three matching functions, respectively. The policy of the welfare state is (η, Ω) = (0.7, 24).
allows for no duration dependence in a sense that would allow hazard rates to fall during an
unemployment spell for a given unemployed worker. In contrast to Ljungqvist and Sargent
(2005), who assume additional probabilistic skill losses while workers are unemployed and
also probabilistic transitions between age classes, the state vector of an unemployed worker
in our matching models is unchanged over the unemployment spell. That implies a constant
hazard rate for a given unemployed worker. So the economy-wide hazard rate in figure 13
falls with the duration of spells in the welfare state because the least employable workers,
those with low but constant hazard rates, constitute an ever larger share of the remaining
unemployed at longer spells. These least employable workers are the low-skilled unemployed
entitled to high benefits who have a low but positive hazard rate in the model with one
matching function, and a zero hazard rate in the model with three matching functions.
7.3
Synthetic Beveridge curves
A time series scatter plot of ū and v is called a Beveridge curve. Because we have analyzed
only stationary equilibria of matching models with time-invariant fundamentals, we cannot
deduce true Beveridge curves from our calculations. However, we can compute something
akin to a Beveridge curve by varying the turbulence parameter π d , while holding other
parameters constant, thereby tracing out equilibrium (ū, v) pairs . Figures 14 and 15 show
the combinations of stationary equilibrium (ū, v) pairs traced out as we vary π d for models
36
0.05
#1
Vacancy rate
0.045
#3
# 2 (current
skills)
0.04
0.035
# 2 (benefits)
0.03
0.025
0.04
0.05
0.06
0.07
0.08
Unemployment rate
0.09
0.1
Figure 14: (Matching model) Synthetic Beveridge curves for steady states of welfare state
(η, Ω) = (0.7, 24) economies with one, two, and three matching functions. Differences in
turbulence π d induce different steady state (ū, v) pairs.
0.05
#1
0.045
0.04
# 2 (current
skills)
Vacancy rate
0.035
0.03
# 2 (benefits)
0.025
0.02
#3
0.015
0.01
0.05
0.1
0.15
0.2
0.25
0.3
Unemployment rate
0.35
0.4
0.45
0.5
Figure 15: (Matching model) Synthetic Beveridge curves for welfare state economies for the
entire range of π d ∈ [0, 1], and consequent bigger ranges of ū.
37
with one, two, and three matching functions. It is important to note that the same variation
in π d leads to different amounts of variation in (ū, v), so that the variation in π d associated
with given movements along these curves differs across the models. For example, we move
π d over the entire range [0, 1] to trace out the curve for the model with one matching model
in figure 14; but variations of π d over a smaller interval [0, π̄ d < 1] are used to trace out that
for the multiple-matching function models. To make this point in a another way, in figure
15 we show the outcomes of varying π d over the entire range [0, 1] for all of the matching
models.
Qualitatively, our synthetic Beveridge curves resemble real ones, sloping downward.
8
Quantitative findings in the search-island model
This section shows that laissez faire and welfare state versions of our search-island model
exhibit the same responses of labor market outcomes as corresponding laissez faire and welfare state economies for the McCall search environment analyzed by Ljungqvist and Sargent
(1998, 2005). Thus, forces neglected by the McCall model but present in the search-island
model (i.e., risk-aversion, precautionary savings, firms that hire capital, and wages determined in competitive markets) fail to blunt the main force captured by the McCall model:
how the incentives for unemployed workers to search change with increasing turbulence. By
worsening the effective skill accumulation technology confronting workers, increased turbulence affects the relative returns to searching and collecting unemployment benefits. Though
workers’ choices of search intensity now also depend on their financial assets and the curvature of their utility function, considerations that are absent from the search model, these
considerations fail to alter the pattern of outcomes.
Figures 16–20 show outcomes in our calibrated search-island model. For zero turbulence,
figures 16 and 17, respectively show that equilibrium unemployment increases with increases
in the UI replacement rate η and that it decreases with increases in the layoff tax Ω, ceteris
paribus. The outcome that layoff taxes suppress unemployment also prevails in the quantitative analysis of Alvarez and Veracierto (2001), who report that equilibrium unemployment
falls by 1.8 percentage points in response to a layoff tax equal to 12 months of wages. In our
search-island model, where workers’s skill levels contribute an additional source of heterogeneity relative to Alvarez and Veracierto’s model, figure 17 shows that unemployment falls
by 0.9 percentage points in response to a layoff tax of 50 which corresponds to roughly 12
months of wages for a low-skilled worker. Alvarez and Veracierto do not study UI replacement rates but instead compute the effects of one-time payments from the government to
laid off workers. Not surprisingly, since they are invariant to the length of unemployment
spells, such severance payments have only a muted (positive) effect on equilibrium unemployment. However, it is interesting to note that Alvarez and Veracierto report that they
assumed a 66% UI replacement rate when initially calibrating their model to U.S. policies
and data, and they found the unemployment rate of that calibrated model to be not much
higher than in the laissez-faire version of their model. This seems puzzling given our figure
16 where the unemployment rate increases dramatically at replacement rates in excess of
38
14
Unemployment rate (per cent)
12
10
8
6
4
2
0
0
0.1
0.2
0.3
0.4
Replacement rate η
0.5
0.6
0.7
Figure 16: (Search model) Unemployment rates for different replacement rates η, given
tranquil economic times and no layoff taxes.
55–60%, but we conjecture that the explanation is that Alvarez and Veracierto assume that
unemployed workers lose their eligibility for unemployment benefits with a constant probability in every period while our unemployed workers keep their benefits throughout their
entire unemployment spells.
As a benchmark parameterization of the welfare state, we set the replacement rate equal
to 0.55 and the layoff tax equal to the above mentioned 12 months of wages for a low-skilled
worker, (η, Ω) = (.55, 50). That yields an equilibrium unemployment rate of 4.1%, which
is lower than the laissez-faire unemployment rate of 4.4%. Thus, both the search-island
model and the matching model can rationalize why unemployment need not be high in the
welfare state in tranquil times. (But as we will soon see, the representative family model
with lotteries cannot rationalize this outcome.) Next, we study how turbulence gives rise to
qualitatively the same effects in the search-island model as in the matching model.
The two panels of figure 18 show the disparate effects of an increase in turbulence on
unemployment rates in the welfare state (η, Ω) = (.55, 50) and laissez-faire (η, Ω) = (0, 0)
economies. The figures show both the total unemployment rate (solid lines) and the percentage of the workers who are unemployed and also had suffered a skill loss after a lay off
in their last job (dashed lines). The dashed lines reveal that the explosion of unemployment
in the welfare state economy when turbulence π d increases is attributable to greater unemployment of previously high-skilled workers who have suffered skill loss upon termination.
In the laissez faire economy, unemployment involving that group increases only mildly with
an increase in turbulence, an outcome that explains why the overall unemployment rate in
the laissez-faire economy is not much affected by increases in turbulence.
The two panels of figure 19 display how the inflow into unemployment and the average
39
5
4.5
Unemployment rate (per cent)
4
3.5
3
2.5
2
1.5
1
0.5
0
0
5
10
15
20
25
30
Layoff tax Ω
35
40
45
50
Figure 17: (Search model) Unemployment rates for different layoff taxes Ω, given tranquil
times and no benefits. The magnitude of the layoff tax can be compared to a semiquarterly
equilibrium wage of 6.4 per unit of skill in the laissez-faire economy, i.e., a layoff tax equal
to 50 corresponds to roughly one year of wage income for a low-skilled worker.
30
5
4.5
25
Unemployment rate (per cent)
Unemployment rate (per cent)
4
20
15
10
3.5
3
2.5
2
1.5
1
5
0.5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
d
Turbulence π
0.7
0.8
0.9
1
(a) Welfare state
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Turbulence πd
0.7
0.8
0.9
1
(b) Laissez-faire economy
Figure 18: (Search model) Unemployment rates in the welfare state (panel a) and the laissezfaire economy (panel b). The solid line is total unemployment. The dashed line shows the
unemployed who have suffered skill loss. The policy of the welfare state is (η, Ω) = (0.55, 50).
40
8
10
Inflow rate (per cent) and duration (quarters)
Inflow rate (per cent) and duration (quarters)
Inflow rate
6
5
4
3
2
Duration
1
0
0
Inflow rate
9
7
8
7
6
5
4
3
2
1
0.1
0.2
0.3
0.4
0.5
0.6
Turbulence
0.7
0.8
0.9
1
(a) Welfare state
0
0
Duration
0.1
0.2
0.3
0.4
0.5
0.6
Turbulence
0.7
0.8
0.9
1
(b) Laissez-faire economy
Figure 19: (Search model) Inflow rate and average duration of unemployment in the welfare
state (panel a) and the laissez-faire economy (panel b). The dashed line is the average
duration of unemployment in quarters. The solid line depicts the quarterly inflow rate
into unemployment as a per cent of the labor force. The policy of the welfare state is
(η, Ω) = (0.55, 50).
duration of unemployment respond to increases in turbulence π d in the welfare state and
laissez-faire economies. In the laissez-faire economy, the inflow rate and the duration are both
impervious to increases in turbulence, while in the welfare state economy the average duration
grows markedly with increases in turbulence; especially at higher levels of turbulence, the
inflow rate into unemployment actually falls modestly with increases in turbulence.
Figure 20 shows how in very turbulent times (π d = 1), hazard rates of gaining employment
behave very differently in the laissez faire and welfare state economies, being flat in the former
and rapidly declining with the length of the unemployment spell in the latter economy.
All of these findings are strikingly similar to the analysis by Ljungqvist and Sargent
(2005), even though that earlier analysis was based on the McCall search environment without many of the features of the search-island model such as risk aversion, precautionary
saving, and competitive firms that hire capital and labor. In turbulent times, the common
feature in these seemingly very different frameworks is the presence of unemployed workers
who have suffered skill loss but are entitled to relatively generous unemployment benefits
based on past earnings. In Ljungqvist and Sargent (2005), these workers were more likely
to become long-term unemployed because they chose relatively high reservation earnings as
compared to their current earnings potential; and given the low likelihood of drawing such
earnings from the wage offer distribution and the mere fact that the generous benefits made
it less costly to stay unemployed, these workers also chose low search intensities. We can
legitimately describe them as “discouraged workers” because they have low probabilities of
41
Laissez faire
Hazard rate of gaining employment (semiquarterly)
1
0.8
0.6
0.4
Welfare state
0.2
0
0
0.5
1
1.5
2
2.5
3
Time (quarters)
3.5
4
4.5
5
Figure 20: (Search model) Semiquarterly hazard rates of gaining employment in turbulent
economic times, π d = 1.0, in the welfare state (dashed line) and in the laissez-faire economy
(solid line). The policy of the welfare state is (η, Ω) = (0.55, 50).
returning to gainful employment. In our search-island model, such workers’ choice of low
search intensities is the only avenue that operates because the abstraction of Alvarez and
Veracierto (2001) has the equilibrium outcome that all workers are paid the same wage rate
per unit of skill. Evidently, this channel by itself is sufficient to explain how unemployment
explodes in the welfare state in response to turbulence while the laissez-faire unemployment
rate remains virtually unchanged.
8.1
Comparing the mechanics of the matching and search-island
models
Even though the search-island and matching models have similar labor market outcomes,
the mechanisms producing those outcomes differ between the two frameworks. As an illustration, consider figures 3 and 16 that show how the unemployment rate is positively
related to the UI replacement rate. The curve for the matching model in figure 3 displays
a more gradual increase, while the corresponding curve for the search-island model in figure
16 is nearly flat for a range of replacement rates below 40–50%. In the matching model,
higher benefits affect the unemployment rate by increasing workers’ threat points for wage
bargaining. As described above, the invisible hand restores equilibrium after an increase in
the replacement rate by lowering market tightness, thereby causing the probability that a
worker encounters a vacancy to fall in order to compensate firms for the lower expected returns from posting vacancies that would occur if market tightness were not lowered. Hence,
all unemployed workers suffer from increased congestion in the labor market. This mecha42
nism by which higher benefits lead to higher unemployment is clearly continuous in the level
of benefits. In contrast, the search-island model features a completely different mechanism
through which higher benefits raise equilibrium unemployment because of the response in
individual workers’ search behavior. Higher benefits make it less costly to remain unemployed, and in response unemployed workers find it optimal to reduce their search intensities
and so lessen the disutility of searching for new employment. But as seen in figure 16, this
effect first becomes significant at relatively high replacement rates. The reason is that the
unemployed workers must fend for themselves. Low replacement rates are of little comfort to
the unemployed who must then finance their consumption with these low benefits and their
savings, so their search intensities are not much affected and the unemployed concentrate
instead on restoring their relatively higher labor market earnings. But at higher replacement
rates, this mechanism for generating unemployment in the search-island model becomes very
potent and workers in effect choose to furlough themselves into drawn out unemployment
spells by setting low search intensities.
9
Quantitative findings in the representative family
model with lotteries
This section shows that the representative family model with employment lotteries reproduces the responses to high UI and high turbulence found by Ljungqvist and Sargent (1998),
but not the response to higher layoff taxes featured in Ljungqvist and Sargent (2005). Figure
21 shows that in tranquil times and with no layoff taxes, increasing the UI replacement rate
η has a powerful effect of increasing the unemployment rate. This outcome agrees with those
from the matching models and the search-island model, but the mechanism is different. Here
there are neither the matching models’ congestion effects nor the search-island model’s search
costs. Therefore, the change in unemployment is not an adjustment of the level of frictional
unemployment that serves to alter firms’ waiting times, thereby compensating them for their
costs of posting vacancies, as in the matching models. Nor does it reflect a longer duration
of unemployment coming from workers’ diminished search activity, as in the search-island
model. Instead, a higher UI replacement rate just makes leisure a more attractive use of the
family’s time.
Figure 22 reaffirms the findings of Hopenhayn and Rogerson (1993) and Ljungqvist (2002)
that with zero turbulence in a representative family model with lotteries, an increase in the
layoff tax causes the unemployment rate to rise, reversing the outcomes in Ljungqvist and
Sargent (2005) and in the matching and search-island models of this paper. In the representative family lotteries model, anyone the family wants to put to work gets a job instantaneously,
so the layoff tax does not have the effect of suppressing frictional unemployment that it does
in our other models. Instead, an increase in the layoff tax decreases the equilibrium wage,
prompting the household to substitute toward leisure.
The model’s sensitivity to UI prompts us to choose a relatively low replacement rate,
η = 0.2, in the benchmark parameterization of the welfare state. We do this to avoid
43
35
Unemployment rate (per cent)
30
25
20
15
10
5
0
0
0.05
0.1
0.15
0.2
Replacement rate η
0.25
0.3
0.35
Figure 21: (Representative family) Unemployment rates for different replacement rates η,
given tranquil economic times and no layoff taxes.
8
7
Unemployment rate (per cent)
6
5
4
3
2
1
0
0
5
10
15
20
25
30
Layoff tax Ω
35
40
45
50
Figure 22: (Representative family) Unemployment rates for different layoff taxes Ω, given
tranquil times and no benefits. The magnitude of the layoff tax can be compared to a
semiquarterly equilibrium wage of 6 per unit of skill in the laissez-faire economy, i.e., a layoff
tax equal to 48 corresponds to one year of wage income for a low-skilled worker.
44
6
40
35
5
Unemployment rate (per cent)
Unemployment rate (per cent)
30
25
20
15
4
3
2
10
1
5
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Turbulence
0.7
0.8
0.9
1
(a) Welfare state
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Turbulence
0.7
0.8
0.9
1
(b) Laissez-faire economy
Figure 23: (Representative family) Unemployment rates in the welfare state (panel a) and
the laissez-faire economy (panel b). The solid line is total unemployment. In the welfare
state, the policy is (η, Ω) = (0.2, 0) and the dashed line shows the unemployed who have
suffered skill loss (which is not a uniquely determined quantity in the laissez-faire economy
and is therefore left out from panel b).
outrageously high unemployment in tranquil times. For the same reason, we set the layoff
tax Ω = 0, since it does not have the effect of suppressing unemployment that it does
in the matching and search-island models. We thus admit defeat in our effort to use the
representative family model in an explanation for how welfare states can sustain relatively
low unemployment rates when π d = 0, and proceed to ask whether the model predicts
that unemployment erupts in the welfare state but not in the laissez-faire economy when
turbulence π d increases. Figure 23 provides an affirmative answer to this question: the
two panels show how an increase in turbulence has very different effects in the welfare
state (η, Ω) = (.2, 0) and the laissez faire (η, Ω) = (.0, 0) versions of this model. In the
welfare state, unemployment increases with turbulence until turbulence reaches about .5,
then unemployment falls slightly with further increases, while under laissez faire, increases
in turbulence push unemployment down to zero.
To shed light on why the laissez-faire unemployment rate falls to zero as turbulence
increases in the right panel of figure 23, it is useful to consider a simplified version of our
representative family model that abstracts from retirement, startup costs for new firms, and
shocks to firms’ productivity. Let z denote a deterministic productivity level of all firms,
which we assume takes a value that induces the family to choose an interior solution for
employment. Given π u > 0 and π d > 0, the family chooses strictly positive steady-state
fractions U , N0 , NH of family members who are unemployed (with low skills), employed
with low skills, and employed with high skills, respectively. The representative family’s Euler
45
equations and the equilibrium conditions that state that the family supplies the economy’s
labor and holds the aggregate capital stock imply the following steady-state values for k and
n:
1
i + δ α−1
k̄ =
[N0 + (1 + H)NH ] ,
αz
(1 − α)(i + δ)(π d + π u ) 1 + βπ u (1 + H) − β(1 − π d )
,
n̄ = N0 + NH =
[i + δ(1 − α)] π d + (1 + H)π u 1 + βπ u − β(1 − π d ) B
where i = (β −1 − 1) is the stationary net interest rate. Note that employment n̄ does not
depend on the productivity level z, as in real business cycle specifications with logarithmic
preferences. But the employment effect of an increase in turbulence is strictly positive and
given by
(1 − α)(i + δ)π u (1 − β) 1 − β + 2β(π u + π d ) + βπ u H
∂ n̄
=
2 2 H > 0 .
∂π d
[i + δ(1 − α)] π d + (1 + H)π u 1 + βπ u − β(1 − π d ) B
Changes in the parameter π d give rise to an equilibrium response in the representative
family’s behavior that can be understood in terms of substitution and wealth effects. An
increase in π d means that the return to working falls, which should reduce the family’s labor
supply because of the substitution effect, on one hand, and increase the family’s labor supply
due to the wealth effect from lower labor income, on the other hand. Evidently, the latter
effect is stronger, since unemployment falls in response to increases in turbulence.
To understand the unemployment effects of increased turbulence in the welfare state,
we divide the range of turbulence in the left panel of figure 23 into three regions; (a) the
positive but relatively flat segment in region π d ∈ [0, .2], (b) the dramatic surge in region
π d ∈ [.2, .5], and (c) the mildly downward-sloping segment in region π d ∈ [.5, 1]. Recall that
our specification of the skill deterioration technology (2) implies that at zero turbulence,
there are no low-skilled workers entitled to high benefits. The only low-skilled workers are
the newly born or workers who have not yet experienced a skill upgrade. At zero turbulence,
the family decides to furlough enough of these workers into leisure that nearly 19% of the
working-age population is unemployed.35
For low levels of turbulence, workers who suffer skill loss after layoffs constitute intramarginal workers in the family’s leisure decision. It is trivially optimal to let these workers
be the first to specialize in leisure because of their higher benefits while they share the same
potential for future skill accumulation as the low-skilled workers who are entitled to low
benefits; they remain intramarginal workers in that leisure decision so long as the family
chooses unemployment for some of the latter workers. Hence, the family’s marginal condition for assigning an additional worker to specialize in leisure at low levels of turbulence
35
As discussed in appendix C.1, we have restricted parameters to guarantee that benefit policies are not
so generous that they would induce families to accumulate skills simply to furlough high-skilled workers into
unemployment and then forgo earning wages in order to collect UI. Hence, there is no steady state with
unemployment among high-skilled workers.
46
pertains to a low-skilled worker who is entitled to low benefits: all low-skilled workers with
high benefits have already been furloughed into unemployment. This marginal condition,
which pertains to the entire region (a), yields an almost constant unemployment rate of
approximately 19%. Thus, as turbulence π d rises until it approaches .2, the family leaves
the total amount leisure roughly constant and just changes the mix of types of workers who
enjoy leisure. Low-skilled workers with low benefits are replaced by low-skilled workers with
high benefits whom the family furloughs into permanent leisure for the rest of their lives.
The slightly positive relationship between turbulence and unemployment in region (a) can
be attributed to an increasing tax wedge on labor supply in response to the government’s
need to balance a budget with higher benefit expenditures (since the number of unemployed
people who collect high benefits increases with turbulence). The higher tax rate on labor
has only a substitution effect because the wealth effect is neutralized when the government
pays out the tax proceeds as unemployment benefits to the representative family.
At a critical level of turbulence of approximately π d = 0.2, the marginal condition for
assigning a low-skilled worker who is entitled to low benefits to specialize in leisure holds
with equality when actually no one of that category is ordered by the family to enjoy leisure.
By then, turbulence has reached the critical level at which the last available unemployed lowskilled worker with low benefits has been replaced by a low-skilled worker entitled to high
benefits – turbulence has entered region (b). Since the marginal condition holds with equality
at that critical level of turbulence, it follows that the family’s first-order condition for letting
an additional low-skilled worker with high benefits specialize in leisure is a strict inequality
when a further increase in turbulence causes that additional worker to ‘materialize’. Thus,
for levels of turbulence in region (b), the family strictly prefers to furlough into leisure
all low-skilled workers who are entitled to high benefits but to furlough none of the lowskilled workers who are entitled to low benefits. As a result, the unemployment rate surges
dramatically with increases in turbulence in region (b) because unemployment increases then
one-for-one with the growing number of skill losers whom the family furloughs into leisure
for the rest of their lives.
When turbulence reaches another critical level of approximately π d = 0.5, the family’s
marginal condition for assigning a low-skilled worker who is entitled to high benefits to
specialize in leisure starts to hold with equality – turbulence has entered region (c). The
representative family sends now some of the skill losers back to work, and it forgoes their
high benefits in exchange for their after-tax wages and their potential for renewed skill
accumulation. In region (c), unemployment falls mildly with increases in turbulence because
of the wealth effect from turbulence, an effect that was described in the case of the laissezfaire economy above. But one might then ask why the equilibrium response to the wealth
effect is so meager in the welfare state as compared to the laissez-faire economy in the right
panel of figure 23, where unemployment plummets in response to turbulence. The reason is
that the representative family of the welfare state faces a very different opportunity set in
which the labor income tax needed to finance the UI system is about 11.4% in region (c),
and the unemployed workers collect an effective replacement rate of 40% on current earnings
potential rather than the stipulated rate of 20%. (Recall that the stipulated replacement rate
47
η = 0.2 is applied to their past earnings, which are twice as high as their current earnings
potential.) It follows that unemployment remains a relatively attractive option for the family
in the welfare state as compared to the family in the laissez-faire economy in which neither
taxes nor benefits distort its labor supply.
9.1
No frictional unemployment and high labor supply elasticity
The employment effects of turbulence in the laissez-faire economy in the right panel of figure
23 are reminders of the fact that the representative family model with employment lotteries
does not embody frictional unemployment. While both the matching and search-island
models generate unemployment because of labor market frictions, there is idleness in the
frictionless representative family model only if the family finds it optimal to let some of its
members specialize in leisure. Even though our preference specification is the same as that
in many real business cycle analyses such as Hansen (1985), we have apparently lost the
standard result that changes in productivity do not affect the steady-state employment to
population ratio. The reason is that we have expanded the notion of productivity beyond
the standard multiplicative productivity level in the aggregate production function to entail
a skill technology in which work experience can enlarge individual workers’ productivities.
This technology introduces new intertemporal considerations into the representative family’s
labor supply decision that prevent substitution and wealth effects in response to steadystate changes in the productivity of that technology from cancelling each other. This makes
possible the outcome that unemployment vanishes in our representative family model when
turbulence increases – an outcome that is not possible in our matching and search-island
models because these models embody frictional unemployment associated with the entry of
new workers into the labor market and the churning of old workers between jobs.
The response of the representative family model to UI stems from the high labor supply
elasticity of that framework, a feature that Hansen (1985) and Rogerson (1988) invoked to
allow the model to explain business cycle fluctuations. That high labor supply elasticity that
apparently works so well in the business cycle context yields unrealistically large equilibrium
responses to the replacement rate in the UI system, a finding that perhaps explains why
earlier researchers have refrained from incorporating European-style generous UI benefits
into the representative family employment lotteries model.
10
Concluding remarks
Our three frameworks for labor market analysis differ: two incomplete markets models –
the matching and search-island models – incorporate different labor market frictions, and
the representative family employment lotteries model has none. Yet except for the positive
response to more EP in the representative family model, the responses to more EP, increased
UI, and increased turbulence are similar in the three frameworks. Though the detailed
mechanisms through which UI and EP affect equilibrium unemployment differ, some common
forces pull all three frameworks. In particular, adverse effects of high UI that are ignited
48
by high turbulence predominate in all three. In subsection 10.1, we highlight the different
mechanisms at work in the three models. In subsection 10.2, we comment on why the
hypothesis advocated by Ljungqvist and Sargent (1998) is so robust that it comes through
in three such different frameworks. In subsection 10.3, we summarize the ways the three
frameworks teach us to think about unemployed European workers and offer our own views
in subsection 10.4 about which description of the situation of those workers is most realistic.
10.1
Comparison of mechanisms
The three models use very different arrangements to match workers with jobs. This leads
them to promote different ways of thinking about how unemployment is determined and
what ‘activity’ it is that unemployed workers do.
1. In the matching models, postulated matching functions impose frictions and externalities on labor markets. By increasing the value of a newly matched worker’s outside
option, increases in UI diminish the surplus value of a match at a given initial draw
of match productivity. To sustain firms’ zero-profit condition for creating jobs, the
invisible hand decreases ‘market tightness’, expressed as a ratio of vacancies to unemployment, thereby decreasing the time firms expect to wait until a vacancy is filled.
That increases the time that an unemployed worker waits to land a job. The matching
models thus focus on how the pools of unemployed workers and vacancies adjust to
sustain firms’ zero-profit condition for posting vacancies. Unemployed workers impose
congestion costs on other unemployed workers, and firms that post vacancies impose
congestion costs on other firms that post vacancies.
2. In the search-island model, the important friction is a costly technology that workers
must use to find jobs.Unemployed workers personally bear search costs. They impose
no congestion costs on other unemployed workers. Alterations in UI affect the unemployment rate by altering an unemployed worker’s choice of search intensity, which
determines the statistical distribution of time that a worker is unemployed and through
it the equilibrium unemployment rate. Thus, the search-island model puts the spotlight
on the individual worker’s incentives to find a job.
3. In the representative family lotteries model, persons who are randomly selected to work
face none of the matching or search frictions present in the matching and search-island
models. Everyone selected to work is immediately hired. The representative family
model focuses on how society chooses to allocate leisure among workers in different
skill and benefit entitlement categories. The representative family allocates work and
leisure in a way that is privately efficient by instructing those workers who are entitled
to the best deals from the UI benefit agency to specialize in leisure. In the matching
and search-island models, match and search frictions generate frictional unemployment.
In the representative family model, there is no frictional unemployment, a difference
that accounts for the models’ differing responses to increases in EP.
49
4. In the matching and search-island models, market incompleteness separates individual workers and makes them bear the consequences of their own actions as well as of
the good or bad luck that comes their way in the form of shocks to their skills and
their employers’ productivities. In the representative family lotteries model, a comprehensive family-wide consumption insurance arrangement protects individual workers
from adverse consequences of those shocks. That protection, the employment lottery,
and the associated high aggregate labor supply elasticity have important quantitative
implications about the response to UI.
10.2
Robustness of the turbulence story
We have used our three models to reexamine the finding of Ljungqvist and Sargent (1998)
that the persistent increase in European unemployment since the 1980s can be explained by
how workers choose to respond to increased turbulence, modelled as a particular deterioration of a stochastic skill transition technology, when the government offers unemployment
benefits. The economic force behind our finding is so strong that it comes through in three
other types of general equilibrium model. In each of the matching models, the search-island
model, and the representative family model, high unemployment erupts in a welfare state
with generous benefits when laid off workers are exposed to more turbulence in their prospective earnings, while the unemployment rate in a laissez-faire economy remains unchanged
(or actually decreases in the representative family model). The higher unemployment pool
in a welfare state consists mainly of workers who have suffered sudden losses of immediate
earnings prospects when laid off. The fact that welfare benefits are based on past earnings
marginalizes these workers by leaving them unemployed for long periods of time. In the
matching models, these unemployed workers with UI benefits that are high relative to their
current earnings prospects encounter fewer acceptable matches because a vacancy’s idiosyncratic productivity must be so much higher to yield a wage rate that can compete with the
high UI benefits. If there are multiple matching functions, these workers have to contend
with low market tightness and the associated relative scarcity of vacancies implies long waiting times before an unemployed worker succeeds in matching with a job. In the search-island
model, workers with generous benefits but poor immediate labor market prospects choose
low search intensities (they become ‘discouraged workers’). In the representative family
model, the family optimally allocates its labor resources by assigning laid off workers who
have high benefits and low immediate earnings potential to specialize in leisure.
Ljungqvist and Sargent (2005) showed that within their McCall model with human capital, high layoff taxes can explain why, until the 1970s brought higher turbulence, unemployment rates were actually much lower in Europe than in the United States. Here, we
have shown that their mechanism operates in two of our three alternative frameworks. In
the matching model and the search-island model, layoff taxes delay the reallocation of labor, lock workers into their current employment, and lower frictional unemployment.36 In
36
Myers (1964, p. 180–181), whom we quoted in footnote 9, also conjectured that stringent EP might
explain the low European unemployment rates in the 1950s and 1960s: “One of the differences [between the
50
contrast, in the representative family model, a layoff tax increases the equilibrium unemployment rate. Layoff taxes diminish labor turnover in the representative family model, but
because there are no frictions in the labor market, there is no frictional unemployment to
suppress. Unemployment rises with increases in layoff taxes because those increases reduce
the equilibrium real wage rate. In response to a lower private return to work, the representative family substitutes away from consumption towards leisure by sending a smaller fraction
of its members to work.37
In the matching and search-island models, government supplied UI and EP interact with
turbulence in ways that can explain the European unemployment experience across the
decades. Despite having levels of UI that were already generous in the 1950s and 1960s,
Europe enjoyed low unemployment because high EP suppressed frictional unemployment by
reducing turnover in the labor market. Inflow rates into unemployment were substantially
lower in Europe than in the U.S., but durations of unemployment spells were approximately
the same. The inflow rates into unemployment remained unchanged with the advent of
turbulence in late 70s, but the length of unemployment spells exploded in Europe.
However, we have been unable to tell this story within the representative family lotteries model, first, because EP does not reduce unemployment in tranquil times, and, second,
because generous UI generates too much unemployment in that framework at all levels of turbulence. We have also been frustrated in our attempt to use the lotteries to understand the
European unemployment experience in terms of observations on the duration of unemployment spells. The reason is that there are multiple ways to design the employment lotteries
that leave the aggregate allocation unaltered but that imply very different life histories for
workers (see appendix C.7). Only for the middle region of the left panel of figure 23 is the
duration of unemployment spells uniquely defined: in this region, the representative family
furloughs all skill losers into unemployment for the rest of their lives, while no one else is
unemployed. But absorbing states of unemployment could also be equilibrium outcomes in
both the left and the right regions. In the left region, the unemployed people with low skills
and low benefits could be newborn workers who have been furloughed into life-time unemployment. In the right region, all unemployed workers have low skills and high benefits, but
since there also exist some workers with those same characteristic who are employed in that
region, there exist multiple lottery designs that support the aggregate allocation, including
U.S. and Europe] lies in our attitude toward layoffs. The typical American employer is not indifferent to the
welfare of his work force, but his relationship to his workers is often rather impersonal. The interests of his
own employers, the stockholders, tend to make him extremely sensitive to profits and to costs. When business
falls off, he soon begins to think of reduction in force . . . In many other industrial countries, specific laws,
collective agreements, or vigorous public opinion protect the workers against layoffs except under the most
critical circumstances. Despite falling demand, the employer counts on retraining his permanent employees.
He is obliged to find work for them to do. . . . These arrangements are certainly effective in holding down
unemployment. But they involve a very heavy cost. They partly explain the traditionally lower productivity
and lower income levels in other countries. Here is something we can learn from our neighbors, therefore,
but are we quite sure we want to learn it? Are there not better ways to reduce unemployment?”
37
For a detailed comparison of the employment implications of layoff taxes in different frameworks, see
Ljungqvist (2002).
51
one where a fraction of workers with these characteristics are furloughed into unemployment
for the rest of their lives, while other skill losers are sent back to employment immediately
after they are laid off. In the laissez-faire economy depicted in the right panel of figure 23,
the only equilibrium restriction is that no high-skilled workers are ever unemployed, so there
is a multiplicity of lottery designs for all levels of turbulence.
10.3
Different views of the European unemployment experience
It is time to think about whether our conversations with the unemployed European workers
in our three models remind us, in Lucas’s words, of ‘talking to an unemployed [European]
friend’.38 Although an adverse interaction between high UI and high turbulence transcends
all three models, the unemployed workers in these models have different alibis for why they
aren’t working.
10.3.1
Unemployed Europeans as winners of a lottery
In response to the question ‘why aren’t you working?’, an unemployed worker in the representative family lotteries model would tell us that he has had the good luck to win a
lottery prize that entitles him to specialize in enjoying leisure.39 In response to ‘but how will
you eat?’, the unemployed worker would tell us that the Europe he lives in functions like
one big happy European-wide family that awards everyone the same consumption stream
independently of his luck in the employment lottery. If he were well read, the unemployed
European worker might detect an Anglo-Saxon drift to our questions and tell us that we
are thinking incorrectly about unemployed European workers. He would remind us that
in his Europe, individual workers are not isolated decision makers who are left to protect
themselves as best they can against the random shocks that economic life throws at them.40
Perhaps, to our surprise, he would credit this outcome to private financial markets – the
complete contingent claims markets that enable sharing the labor income risk associated
with the employment lotteries – not to the activities of his government. Seen from a private
perspective, government-provided UI enriches the collective of workers engaging in the employment lotteries, but seen from a social perspective the associated adverse incentive effects
impoverish the economy. Even modest amounts of government-provided UI threaten to send
the economy down into an abyss.
The private insurance markets in the representative family model make governmentprovided unemployment insurance not only unnecessary but a recipe for disaster. Employment lotteries and contingent claims markets yield a high aggregative labor supply elasticity.
Prescott (2004) vividly demonstrated the potency of that high labor supply elasticity in his
38
We have added ‘European’ to Lucas’s phrase.
If we were to ask him when he had last worked, he might tell us about a job loss and skill transition
event that, according to the lottery ticket he had drawn, declared him a leisure-specializing winner of the
lottery contingent on the history of shocks corresponding to that event. Recall the discussion of figure 23.
40
And he might tell us that Chiappori (1992, 1997), and Browning and Chiappori (1998) mistakenly drew
the boundary of their family around a nuclear instead of a national family.
39
52
time series analysis of the American-European labor market divide. After observing similar
numbers in the early 1970s, Prescott attributes the dramatic decline of 20 to 30 percent in
hours worked per person of working age in Germany and France in the 1990s to tax increases
of 7 and 10 percent, respectively.41 Prescott incorporates neither government-provided UI
nor any other welfare benefits (see footnote 11). In light of the large equilibrium responses
to the tax increases studied by Prescott, think about what would happen if we were to
introduce European-style social insurance into Prescott’s framework: figure 21 shows how
economic activity in the representative agent lotteries model shuts down with generous government supplied UI. Evidently, the high labor supply elasticity in the representative family
lotteries model implies that generous government welfare programs would lead to the ultimate ‘abuse’ of such social insurance. Hence, a seemingly fatal shortcoming of that model
for explaining the European employment experience is that the model predicts far too low
a labor supply for realistically calibrated levels of government-provided UI. With realistic
levels of government-provided UI, the puzzle for that model becomes, why do Europeans
work at all?
10.3.2
Unemployed Europeans as victims of congestion and their own bargaining power
In the matching models, individual unemployed workers are isolated decision makers, not
members of a national representative family. Because they are pooled with other workers
who also wait in the same matching function, their employment prospects and anticipated
durations of unemployment depend on the characteristics of those other workers and how
costly it is for firms to create vacancies. The mixture of skills and UI benefit entitlements of
the workers in a pool determine how profitable it is for firms to post vacancies there. In a
matching function that includes low-skilled workers with different benefits, it is more costly
for firms to match with workers with low skills who are entitled to high UI than to match with
workers with low skills who are entitled to low UI. It requires a higher productivity draw to
form successful matches with the former workers as compared to the latter workers and for a
given productivity draw, bargaining with the former workers leads to higher wage payments
as compared to bargaining with the latter workers. To induce firms to post vacancies in a
pool with many low skill, high UI entitlement workers, the invisible hand must set market
tightness θ = v/u to be low enough that firms can expect to match with workers frequently
enough. But this means that workers can expect to match infrequently with firms, so that
their expected durations of unemployment are high. Thus, if we were to ask an unemployed
worker waiting in one of the matching models why he had been unemployed for so long,
he could blame the other workers in his matching function whose bad characteristics are
responsible for making market tightness low.
If pressed further about the reasons for his long unemployment spell, a low-skilled unem41
Ljungqvist (2005) offers a qualification to this statement by showing that two thirds of the predicted
employment effects in Prescott’s analysis is due to the tax increases while one third is caused by movements
in Prescott’s estimate of a consumption to output ratio that determines the wealth effect of taxation.
53
ployed worker with high benefits would perhaps concede that his tough bargaining posture
during ‘job interviews’ might contribute to his current predicament. Given his high benefits
relative to his current skills, he will confront each prospective employer with a sizable threat
point during the Nash bargaining process. If we were to ask the worker how he expects to
get away with such outrageous wage demands when there are so many other unemployed
workers also waiting in the same matching function, he could point to the fact that the firm
would face him as the sole job candidate. If the firm would like to meet other candidates,
it would have to expend additional resources on posting a vacancy and incur a delay of at
least one more period before meeting someone else. Besides, any future encounters would
also be on a bilateral basis. A firm could never expect to meet more than one job seeker at
the negotiation table.
If the economy consigns low skill, high UI entitlement workers to their own matching
function, it is true that unemployment rates for workers not in that ‘skill-losers market’ will
be lower. But when UI benefits are high and turbulence is high, the existence of such an
isolated market for losers of skills becomes a misfortune for the steady state of the economy,
because high turbulence destines so many workers to pass through the skill-losers market in
which firms choose optimally to post very few vacancies. Hence, the unemployment rate in
such an economy literally explodes when turbulence increases because losers of skills become
victims of long-term unemployment in their overly congested matching function. Having a
single matching function in the economy dilutes the adverse consequences for unemployment
of high turbulence, but even then, by creating more skill losers in the pool of prospective
workers, high turbulence in conjunction with high UI causes equilibrium unemployment to
grow in order to induce firms to post vacancies.42
10.3.3
Unemployed Europeans as discouraged workers
Because it emphasizes the factors that influence an unemployed worker’s choice of search
intensity, talking with an unemployed worker in the search-island model most closely resembles the conversation that Lucas imagined with an unemployed worker in the McCall
model. An individual’s search intensity is the only factor that determines his duration of
unemployment. A worker’s search intensity depends on his asset level, his skill level, his
benefit entitlement, and, as a determinant of the skill accumulation technology, the level
of the turbulence parameter π d . For a given level of turbulence, workers with low skills,
high accumulated financial assets, and high benefit entitlements choose the lowest search
intensities. The weak incentives to search provided by their high UI entitlements and their
42
Mortensen and Pissarides (1999) study a matching model with heterogeneous workers endowed with
different skill levels who match with firms in separate skill-specific matching functions. They show that a
mean-preserving spread in the distribution of the labor force over skill types can explain the divergent labor
market performance in the U.S. and Europe, given differences in UI and EP. Pissarides (1992) analyzes skill
accumulation in a two-period overlapping generations model where all workers match with firms in a single
matching function. His focus is on how the externality in the matching process gives rise to a propagation
mechanism where a temporary shock to employment can persist for a long time.
54
current low skills are what discourages these ‘discouraged workers’.43
The search-island model and the representative family lotteries model have the common
feature that there are no externalities in the labor market. A worker selling his labor services
or a firm buying those services does not inflict injuries on others in the labor market beyond
what a seller or a buyer of a good ordinarily imposes on competitors. Furthermore, since the
wage rate is determined competitively in these two models, the holdup problem present in
the bargaining setting in the matching model is absent. Thus, unemployment in the searchisland model and the representative family lotteries model both directly reflect workers’
decisions. After allowing for the time lag imposed by the time-to-create-jobs assumption,
firms will create a number of new jobs that is adequate for all workers who are willing to
work at the competitive wage, and who have ‘found the labor market’ in the search-island
model.
However, the search-island model shares two crucial features with the matching models:
labor market frictions and incomplete markets. Those features enable these frameworks to
explain why unemployment rates were lower in Europe than in the U.S. until the late 1970s,
in spite of higher UI and more stringent EP in Europe – outcomes that remain a mystery
for the representative family lotteries model.
10.4
Our opinion
We have let the models, and the unemployed workers living inside them, speak for themselves.
It is wiser for the reader to listen to those workers than to us. But for what they are worth,
we state our reactions to what these imaginary workers have told us. While all three models
bear important insights, our conversation with the discouraged worker in the search-island
model of section 10.3.3 rings truest to us. That is why we continue to advocate the extended
McCall framework of Ljungqvist and Sargent (1998, 2005) as a good way to think about the
choices confronting European workers and policy makers. The very low employment rates
reported for the European economy with employment lotteries in section 9 do not capture
the fact that the welfare states of Europe actually provided low unemployment rates before
the 1970s. Viewing outcomes in Europe in the 1950s and 1960s from the point of view of the
quantities that emerge from the search-island and matching models, on the one hand, and
the representative family lotteries model, on the other, would make us think that those good
European outcomes occurred until the 1970s because there were incomplete markets and no
employment lotteries, forcing each household to seek its own fortune either in labor markets
or in government welfare programs. We might say, “thank goodness that Europe has actually
not had the nation-wide private consumption insurance and employment lotteries that in
the presence of realistically calibrated UI benefits would have brought economic activity to
a standstill!”
43
If one could account for the discouraged workers who have taken advantage of early retirement and
disability programs available in Europe, we suspect that, building on the ideas of Edling (2005), one would
find even more discouraged workers than are recorded in the unemployment statistics. See the remarks in
footnote 10.
55
Appendices
Three appendices describe the models in enough detail to prepare computer programs to
compute their equilibria.
A
A.1
Matching models
Single matching function
When there is a single matching function, the probability that a firm meets a worker with
skill h and benefit entitlement b is
λf (h, b) =
u(h, b)
M (v, ū) u(h, b)
= m(θ)
v
ū
v
(12)
and the probability that a worker with skill h and benefit entitlement b is matched with a
vacancy is
M (v, ū)
= m(θ),
(13)
λw (h, b) =
ū
which is independent of (h, b). When we introduce multiple matching functions in subsection
A.5, the probability that a worker with skill h and benefit entitlement b is matched with a
vacancy will depend on (h, b).
A.2
Match surplus
When an unemployed worker with skill h and benefit entitlement b meets a firm with a
vacancy, the firm-worker pair draws productivity z from a distribution Qoh (z). The firm and
the worker will stay together and produce if the match surplus S o (h, z, b) is positive:
n
(1 − τ )z − [1 − β(1 − ρ)] W (h, b)
S o (h, z, b) =
max
{stay, depart}
o
X
n
0
0
0 0
o
o
p (h, h )Qh0 (z, z )S(h , z ) , 0 , (14)
+β(1 − ρ) −π Ω + (1 − π )
h0 ,z 0
where W (h, b) is the worker’s outside value, and S(h, z) is the surplus associated with a
continuing match. A worker with skill h and benefit entitlement b has an outside option
with value
X
w
o
o
W (h, b) = b̃(b) + β(1 − ρ) W (h, b) + λ (h, b)
ψS (h, z, b)Qh (z) .
(15)
z
Free entry makes the firm’s outside value zero. The firm and worker split the match surplus
S o (h, z, b) through Nash bargaining, with outside values as threat points. Let ψ ∈ (0, 1)
denote the worker’s share of the match surplus. Because both parties want a positive match
56
surplus, there is mutual agreement on whether to form a match. The reservation productivity
z̄ o (h, b) satisfies
S o (h, z̄ o (h, b), b) = 0.
(16)
The surplus of a continuing match is
n
S(h, z) =
max
(1 − τ )z − [1 − β(1 − ρ)] W (h, h)
{continue, break up}
o
X
n
0
0
0 0
o
o
p (h, h )Qh0 (z, z )S(h , z ) , −Ω . (17)
+β(1 − ρ) −π Ω + (1 − π )
h0 ,z 0
The government’s policy of imposing a layoff tax Ω on matches that break makes (17) differ
from expression (14).44 A reservation productivity z̄(h) satisfying
S (h, z̄(h)) = −Ω
(18)
characterizes whether a match is dissolved.
A.3
Equilibrium condition
In equilibrium, firms expect to break even when posting a vacancy:
X
µ = β(1 − ψ)
λf (h, b)S o (h, z, b)Qoh (z).
(19)
h,z,b
This condition will pin down the equilibrium value of market tightness θ.
A.4
Wage determination
Alternative wage structures support the same equilibrium allocation. We follow Mortensen
and Pissarides (1999) and assume a two-tier wage system.45 In particular, when a firm with
a vacancy meets an unemployed worker with skill h and benefit entitlement b, they bargain.
The worker’s outside value is W (h, b) and the firm’s outside value is zero. Because they do
not incur the layoff tax if they do not reach an agreement, the layoff tax does not directly
affect the bargaining between a newly matched worker and firm. But if they succeed in
forming a match, the firm must pay the layoff tax after any future breakup. We capture this
by setting the firm’s threat point equal to −Ω in future Nash bargaining.
44
Another difference between expressions (14) and (17) is that an employed worker’s benefit entitlement
is encoded in his skill level h, so there is one less state variable in surplus expression (17).
45
The risk neutral firm and worker would be indifferent between adhering to this two-tier wage system or
one in which workers receive a fraction ψ of the match surplus S(h, z) in every period (which would have
the worker paying a share ψ of any future layoff tax). As emphasized by Ljungqvist (2002), the wage profile,
not the allocation, is affected by the two-tier wage system. Optimal reservation productivities remain the
same. Under the two-tier wage system, a newly hired worker in effect posts a bond that equals his share of
the future layoff tax.
57
These assumptions give rise to a two-tier wage system. There is one wage function
w̃ (h, z, b) for the initial round of negotiations between a newly matched firm and worker,
and another wage function w̃(h, z) associated with renegotiations in an ongoing match. These
wage functions satisfy
X
o
o
po (h, h0 )W (h0 , h)
w̃ (h, z, b) = W (h, b) + ψS (h, z, b) − β(1 − ρ) π o
o
X
h
n
0
0
0 0
0
0
p (h, h )Qh0 (z, z ) ψ S(h , z ) + Ω + W (h , h ) , (20)
X
h
n
0
0
0 0
0
0
p (h, h )Qh0 (z, z ) ψ S(h , z ) + Ω + W (h , h ) . (21)
0
+(1 − π o )
h0 ,z 0
X
w̃(h, z) = W (h, h) + ψ S(h, z) + Ω − β(1 − ρ) π o
po (h, h0 )W (h0 , h)
0
+(1 − π o )
h0 ,z 0
A.5
Multiple matching functions
We entertain some alternative specifications that proliferate matching functions in the spirit
of Mortensen and Pissarides (1999), who postulated that workers with different skill levels
get matched with vacancies in separate but identical matching functions with market-specific
inputs of unemployment and vacancies. We must modify their specification because they
assumed that workers are permanently endowed with a particular skill level, and we don’t.
We consider three alternative specifications:
1. Separate matching functions for unemployed workers with different skill levels, yielding
different equilibrium vacancies v(h) for h ∈ {0, H}.
2. Separate matching functions for unemployed workers having different benefit entitlements, yielding different equilibrium vacancies v(b) for each b ∈ {0, H}.
3. Separate matching functions for unemployed workers indexed by both their current
skill h and their skill b in their last employment, yielding equilibrium vacancies v(h, b)
for each pair of values (h, b) ∈ {0, H} × {0, H}.
Case 1: When workers are sorted according to their current skills h, tightness in market h is
v(h)
.
b u(h, b)
θ(h) = P
58
(22)
The probabilities that an unemployed worker finds a vacancy and that a firm with a vacancy
finds a worker, respectively, equal
P
M v(h), b u(h, b)
P
λw (h, b) =
= m(θ(h)) ,
(23)
b u(h, b)
P
M v(h), b u(h, b)
u(h, b)
u(h, b)
P
= m(θ(h))
.
(24)
λf (h, b) =
v(h)
v(h)
b u(h, b)
The zero-profit condition for posting a vacancy in the market for unemployed workers with
skill h is
X
µ = β(1 − ψ)
λf (h, b)S o (h, z, b)Qoh (z),
(25)
z,b
where µ is the cost of posting a vacancy.
Case 2: When workers are sorted according to their skills b when last employed, the tightness
in market b is
v(b)
.
(26)
θ(b) = P
h u(h, b)
The probabilities that an unemployed worker finds a vacancy and that a firm with a vacancy
finds a worker, respectively, equal
P
M v(b), h u(h, b)
P
λw (h, b) =
= m(θ(b)) ,
(27)
h u(h, b)
P
M v(b), h u(h, b)
u(h, b)
u(h, b)
P
.
(28)
= m(θ(b))
λf (h, b) =
v(b)
v(b)
h u(h, b)
The zero-profit condition for posting a vacancy in the market for unemployed workers whose
skills were b in their last employment becomes
X
µ = β(1 − ψ)
λf (h, b)S o (h, z, b)Qoh (z).
(29)
h,z
Case 3: When workers are sorted both according to their present skills h and their skills b
when last employed, the tightness in each separate market, indexed by (h, b), is given by
θ(h, b) =
v(h, b)
.
u(h, b)
59
(30)
The probabilities that an unemployed worker finds a vacancy and that a firm with a vacancy
finds a worker, respectively, equal
M v(h, b), u(h, b)
λw (h, b) =
= m(θ(h, b)) ,
(31)
u(h, b)
M v(h, b), u(h, b)
1
λf (h, b) =
= m(θ(h, b))
.
(32)
v(h, b)
θ(h, b)
The zero-profit condition for posting a vacancy for unemployed workers with current skill h
and skill b when last employed is
X
µ = β(1 − ψ)λf (h, b)
S o (h, z, b)Qoh (z).
(33)
z
B
B.1
Search-island model
Firm’s problem
The Bellman equations of an existing firm are
o
n
f
f
V (h, z) = max Ṽ (h, z) , −Ω ,
1−α
f
α
∗
Ṽ (h, z) = max zk (1 + h)
− w (1 + h) − (i + δ) k
k
X
1−ρ
o
o
n
0
f
0 0
0
+
−π Ω + (1 − π )
p (h, h ) V (h , z ) Q (z, z ) .
1+i
0
0
h ,z
(34)
(35)
The first-order condition for capital in problem (35) is
zαk α−1 (1 + h)1−α = (i + δ) ,
(36)
which can be solved for k to obtain the firm’s policy function for choosing capital,
zα
k (h, z) =
i+δ
1
1−α
(1 + h) .
(37)
Associated with the solution to an existing firm’s optimization problem is a reservation
productivity z̄(h) that satisfies
Ṽ f (h, z̄(h)) = −Ω.
(38)
Define the following indicator function
Λ (h, z) =
if z ≥ z̄(h);
otherwise.
1,
0,
60
(39)
The break-even condition for starting a new firm is
n
o
1 X
f
f
max (1 − φ)Ṽ (0, z) + φṼ (H, z) , 0 Qo (z) ,
µ=
1+i z
(40)
where µ is the start-up cost and φ is the fraction of high-skilled workers among all new hires.
The maximization in (40) implies a reservation productivity z̄ o that determines whether a
new firm hires a worker after it observes its productivity level. The reservation productivity
satisfies
(1 − φ)Ṽ f (0, z̄ o ) + φṼ f (H, z̄ o ) = 0.
(41)
Define the following indicator function
o
Λ (z) =
if z ≥ z̄ o ;
otherwise.
1,
0,
(42)
The productivity distribution of new firms that hire workers is
Λo (z) Qo (z)
.
o 0
o 0
z 0 Λ (z ) Q (z )
B.2
Γ(z) = P
(43)
Household’s problem
We define three value functions V n (a, h, z), V u (a, h, b), and V r (a) for an employed worker, an
unemployed worker, and a retired worker, respectively. The state variables are last period’s
assets (a), skill index (h), the firm’s current productivity level if employed (z), and the
worker’s benefit entitlement if unemployed (b). The benefit entitlement is determined by the
worker’s last earnings, which we index by b ∈ {0, H}, his skill index when he last worked.
Both newborn unemployed workers and laid off unskilled workers have a benefit entitlement
indicated by index b = 0.
The Bellman equation of an employed worker is
X
r 0
n
po (h, h0 ) V u (a0 , h0 , h)
log c + βρV (a ) + β(1 − ρ) π o
V (a, h, z) = max
0
c,a
h
n
pn (h, h0 ) V n (a0 , h0 , z 0 ) Λ (h0 , z 0 )
0
+(1 − π o )
X
h0 ,z 0
o
0
+V (a , h , h) [1 − Λ (h , z )] Q (z, z )
u
0
0
subject to
c + a0 ≤ (1 + i) a + (1 + h) w ,
c, a0 ≥ 0 .
61
0
0
(44)
Policy functions c̄n (a, h, z) and ān (a, h, z) give the employed worker’s optimal levels of consumption and savings, respectively.
The Bellman equation of an unemployed worker is
(1 − s)γ − 1
u
V (a, h, b) = max
log
c
+
A
+ βρV r (a0 ) + β(1 − ρ)
c,a0 ,s
γ
X
u 0
ξ
ξ
n 0
0
0
· 1 − s V (a , h, b) + s
V (a , h, z ) Γ (z )
(45)
z0
subject to
c + a0 ≤ (1 + i) a + η (1 + b) w ,
c, a0 ≥ 0 , s ∈ [0, 1).
Policy functions c̄u (a, h, b), āu (a, h, b), and s̄ (a, h, b) give the unemployed worker’s optimal
levels of consumption, savings, and search effort, respectively.
The Bellman equation of a retired worker is
h
i
r 0
V r (a) = max
log
c
+
β(1
−
σ)V
(a
)
(46)
0
c,a
subject to
c + a0 ≤ (1 + i) a ,
c, a0 ≥ 0 .
Policy functions, c̄r (a) and ār (a), give optimal consumption and savings, respectively.
B.3
Steady state
In a steady state, a time-invariant measure N (h, z) describes the number of firms operating
with workers of skill index h ∈ {0, H} and productivity level z. This measure must be
consistent with the stochastic process for idiosyncratic shocks and the employment decisions
of firms. If v is the number of newly created firms, then N (· , ·) must satisfy
N (0, z 0 ) = vQo(z 0 )Λo(z 0 )(1 − φ) + (1 − ρ)(1 − π o )Λ(0, z 0 )
X
·
pn(h, 0)N (h, z)Q(z, z 0 ),
(47)
h,z
0
N (H, z ) = vQ (z )Λ (z )φ + (1 − ρ)(1 − π o )Λ(H, z 0 )
X
·
pn(h, H)N (h, z)Q(z, z 0 ).
o
0
o
0
(48)
h,z
Time-invariant measures y n (a, h, z), y u (a, h, b), and y r (a), respectively, describe the numbers of employed, unemployed, and retired households with various individual characteristics.
62
These measures are implied by the optimal decision rules by firms and households:
X
n 0
0 0
y (a , h , z ) = (1 − ρ) (1 − π o )Λ (h0 , z 0 )
pn (h, h0 ) y n (a, h, z) Q (z, z 0 )
a,h,z:ān(a,h,z)=a0
X
0
+Γ (z )
a,b:āu(a,h0 ,b)=a0
(
X
y u (a0 , h, b) = (1 − ρ) π o
ξ
u
0
(49)
po (b, h) y n (a, b, z)
a,z:ān(a,b,z)=a0
X
+(1 − π o )
a,z,z 0 :ān(a,b,z)=a0
+
s̄(a, h , b) y (a, h , b) ;
0
X
h
i
pn (b, h) y n (a, b, z) 1 − Λ (h, z 0 ) Q (z, z 0 )
h
y u (a, h, b) 1 − s̄ (a, h, b)ξ
a:āu(a,h,b)=a0
X
+I(h, b) σ
i
)
y r (a) ,
(50)
a:ār(a)=a0
y r (a0 ) = (1 − σ)
X
y r (a)
a:ār(a)=a0
+ρ
X
n
y (a, h, z) +
a,h,z:ān(a,h,z)=a0
X
u
y (a, h, b) ,
a,h,b:āu(a,h,b)=a0
(51)
where I(h, b) is an indicator function that equals one if h = b = 0 and zero otherwise.
Following Alvarez and Veracierto (2001),we consider steady-state equilibria without public debt. The government balances its budget every period, implying
X
X
0 = (w∗ − w)
(1 + h)N (h, z) + ΩD − ηw
(1 + b)y u (a, h, b) ,
(52)
h,z
a,h,b
where the amount of job destruction D is
X
h
i
X
o
n
0
0 0
0
o
N (h, z) + (1 − π )
p (h, h ) 1 − Λ (h , z ) N (h, z) Q (z, z ) . (53)
D = (1 − ρ) π
h,z
h,h0 ,z,z 0
The market-clearing condition in the goods market is
X
c̄ + δ k̄ + µv =
N (h, z) z k(h, z)α (1 + h)1−α ,
(54)
h,z
where aggregate consumption and the aggregate capital stock, respectively, are
X
X
X
c̄ =
c̄n (a, h, z) y n (a, h, z) +
c̄u (a, h, b) y u (a, h, b) +
c̄r (a) y r (a) ,
a,h,z
k̄ =
X
a,h,b
N (h, z) k(h, z).
(55)
a
(56)
h,z
63
There are two equilibrium
in the labor market. First, the measure of new
P conditions
o
o
firms that hire workers, v z Λ (z)Q (z), must equal the measure of unemployed workers
who accept employment. Second, the skill ratio φ among new hires that the firm takes as
exogenous must equal the equilibrium skill ratio among new hires. We can use the timeinvariant population measures to express these equilibrium conditions as:
X
(1 − ρ)
s̄(a, h, b)ξ y u (a, h, b)
a,h,b
v =
X
,
Λo (z) Qo (z)
(57)
z
φ =
X
s̄(a, H, b)ξ y u (a, H, b)
a,b
X
s̄(a, h, b)ξ y u (a, h, b)
.
(58)
a,h,b
Households’ aggregate demand for assets
X
X
X
ā =
a y n (a, h, z) +
a y u (a, h, b) +
a y r (a) ,
a,h,z
a,h,b
(59)
a
should equal the supply of assets, which consists of the aggregate capital stock k̄ and the
value of claims to the economy’s firms:
i
P h
α
1−α
∗
− w (1 + h) − (i + δ)k(h, z) N (h, z) − µv − ΩD
h,z z k(h, z) (1 + h)
ā = k̄ +
. (60)
i
C
C.1
Representative family model with lotteries
Permissible benefit policies
We assume benefit policies that are not so generous that they would induce families to
accumulate skills simply to furlough high-skilled workers into unemployment and then forgo
earning wages in order to acquire benefits from the public sector. This assumption implicitly
generates a restriction on benefit policies that can be derived by taking a steady state in which
the family initially enjoys leisure by keeping some of its low-skilled workers unemployed,
then asking how the family’s wealth would change were it to send an unemployed low-skilled
family member to work with the intention of furloughing him into unemployment after he
has attained the higher skill level. We impose that during the skill accumulation phase for
that worker, the family keeps its leisure unchanged by temporarily furloughing an already
high-skilled worker into unemployment. This strategy gives rise to stochastic streams of
costs during the worker’s skill accumulation phase and payoffs after the worker has attained
the higher skill level. These can be exchanged for their expected present values evaluated at
a stationary interest rate equal to (1 + i) = β −1 .
64
During the accumulation phase, when the low-skilled worker replaces the high-skilled
worker in the labor market, the family gains an amount (1 − η)w per period from sending
the low-skilled worker to work but loses an amount (1 − η)(1 + H)w from furloughing the
high-skilled worker into unemployment. Thus, the impact on the family’s disposable income
during the accumulation phase is −(1 − η)Hw per period. This loss continues for another
period with probability (1−π u ), i.e., so long as the low-skilled worker does not experience an
upgrade in skills.46 But with probability π u , the low-skilled worker attains the higher skill
level. When that happens, the family sends the originally high-skilled worker back to work
and furloughs the originally low-skilled but newly high-skilled worker into unemployment.
That originally low-skilled worker is now entitled to benefits that exceed his earlier benefit
level by ηHw. The family keeps this stream of a higher disposable income until the worker
with the newly upgraded skill level retires.
H
Let κH
0 be the capitalized value of this whole strategy on its inception, and let κH be the
capitalized value of the higher benefit stream at the time when the low-skilled worker gains
the higher skill level and is furloughed into unemployment. These capitalized values satisfy
i
h
u H
u H
(61)
κH
=
−(1
−
η)Hw
+
β
π
κ
+
(1
−
π
)κ
0
H
0 ,
H
κH
H = ηHw + β(1 − ρ)κH .
(62)
After solving for κH
H from equation (62) and substituting into equation (61), we can solve
for the capitalized value associated with this strategy,
βπ u η
1 − β(1 − ρ)
Hw.
1 − β(1 − π u )
−(1 − η) +
κH
0 =
(63)
We require that a permissible benefit policy make this strategy unprofitable, so that κH
0 ≤ 0.
This implies that
βπ u η ≤ [1 − β(1 − ρ)](1 − η).
(64)
This condition implies an upper bound on the generosity of the replacement rate η. Alternatively, for a given replacement rate η, expression (64) states that the probability π u of
experiencing an upgrade and the subjective discount factor β together must be sufficiently
low that it is not worthwhile to accumulate skills just in order to collect benefits at the higher
skill level. Thus, we set benefit levels so that it is in the representative family’s interest to
reap the returns from any skill accumulation that come from working.
C.2
Steady-state employment and population dynamics
We study an economy in a stochastic steady state. A representative family runs the household
sector. In a steady state, the family’s optimal policy is characterized by two flow rates into
46
The retirement probability ρ does not enter these calculations, because if either the low-skilled or the
high-skilled worker retires while the strategy is being executed, the family will just replace that worker with
another person from his skill category.
65
unemployment: a fraction e0 of newborns that enter life-time unemployment, and a fraction
e4 of all laid off workers with skill losses who enter unemployment for the rest of their
lives.47 When the benefit policy satisfies restriction (64), there is no unemployment among
high-skilled workers.
At time t, let Rt be the fraction of a family’s members who are retired. The remaining
working-age members are divided into four categories. Let U0t , U4t , N0t , and NHt be the
fractions of a family’s members who are unemployed from birth, unemployed after suffering skill loss, employed with low skills, and employed with high skills, respectively. These
fractions satisfy
Rt + U0t + U4t + N0t + NHt = 1.
(65)
For given flow rates (e0 , e4 ), the laws of motion are
i
h
Rt = (1 − σ)Rt−1 + ρ U0t−1 + U4t−1 + N0t−1 + NHt−1 ,
U0t = (1 − ρ)U0t−1 + e0 σRt−1 ,
h
i
o d
U4t = (1 − ρ) U4t−1 + π π e4 NHt−1 ,
nh
i
o
N0t = (1 − ρ) 1 − (1 − π o )π u N0t−1 + π o π d (1 − e4 )NHt−1 +(1 − e0 )σRt−1 ,
nh
i
o
NHt = (1 − ρ) 1 − π o π d NHt−1 + (1 − π o )π u N0t−1 .
(66)
(67)
(68)
(69)
(70)
We can use equations (65) and (66) to solve for the stationary fraction of retired members
ρ
,
σ+ρ
R=
(71)
which can be substituted into equation (67) to obtain the stationary fraction of family
members who have been unemployed since birth
U0 =
e0 σ
.
σ+ρ
(72)
To compute the stationary labor allocation, we start with equation (70) and express NH in
terms of N0 ,
(1 − ρ)(1 − π o )π u
NH =
N0 ,
(73)
1 − (1 − ρ)(1 − π o π d )
which can be substituted together with equation (71) into equation (69) and then solved for
h
i
1 − (1 − ρ)(1 − π o π d ) (1 − e0 )σρ
N0 =
,
(74)
χe (σ + ρ)
47
These flow rates into unemployment correspond to one particular design of the employment lottery, but
there are many other designs that implement the same steady-state aggregate labor allocation and yield the
same expected utility to workers. See section C.7.
66
where
n
h
i
o
o d
o
u
o d
o
u
χ ≡ 1 − (1 − ρ) 1 + ρ 1 − π π − (1 − π )π − (1 − ρ)π π e4 (1 − π )π > 0;
e
(75)
χe is strictly positive since
n
h
io
χe ≥ 1 − (1 − ρ) 1 + ρ 1 − π o π d − (1 − π o )π u ≥ 1 − (1 − ρ)(1 + ρ) = ρ2 > 0.
By using equations (73) and (74), we can solve for U4 from equation (68),
U4 =
(1 − ρ)2 π o π d e4 (1 − π o )π u (1 − e0 )σ
.
χe (σ + ρ)
(76)
It is interesting to note that the skill composition of employed workers is a function
only of exogenous parameters and does not depend on the choice of flow rates (e0 , e4 ). Use
equation (73) to compute
φN =
C.3
NH
(1 − ρ)(1 − π o )π u
=
∈ (0, 1).
N0 + NH
ρ + (1 − ρ)π o π d + (1 − ρ)(1 − π o )π u
(77)
A perturbation of employment
Before turning to equilibrium labor dynamics in a steady state, we examine two perturbations
from a steady-state labor allocation. We will use these perturbations to compute a steady
state.
Suppose that the steady state is such that the representative family has a positive measure of unemployed workers who have suffered a skill loss. We can then ask: how would
the family’s wealth change if the set of unemployed workers who have suffered skill loss is
permanently reduced by one worker? That is, the family considers sending one such worker
to the labor market and, when he retires, replacing him with another unemployed worker
who has suffered skill loss. Such a succession of workers will give rise to a stochastic stream
of labor income that the family can immediately exchange for the expected present value of
the stream discounted at the stationary interest rate (1 + i) = β −1 .
Let κ4
0 be the capitalized value of the labor income associated with this strategy of
reducing unemployment among workers who have suffered skill loss. Let κ4
h be the capitalized
value of the stream of labor income at a future time when this worker (or one of his successors)
has attained high skills. These capitalized values satisfy
nh
i
o
4
4
o
u
o
u 4
κ0 = w + β(1 − ρ) 1 − (1 − π )π κ0 + (1 − π )π κH + βρκ4
(78)
0 ,
o
n
o d 4
o d 4
+ βρκ4
(79)
κ4
0 ,
H = (1 + H)w + β(1 − ρ) (1 − π π )κH + π π κ0
where w is the market-clearing after-tax wage rate.
67
We can use equation (79) to solve for κ4
H,
h
i
(1 + H)w + β (1 − ρ)π o π d + ρ κ4
0
,
κ4
H =
1 − β(1 − ρ)(1 − π o π d )
(80)
which can be substituted into equation (78),
κ4
0 =
1 − β(1 − ρ)(1 − π o π d ) + β(1 − ρ)(1 − π o )π u (1 + H)
w > 0,
χ0
(81)
where
in
h
i
o
1 − β(1 − ρ)(1 − π o π d ) 1 − β(1 − ρ) 1 − (1 − π o )π u − βρ
h
i
−β(1 − ρ)(1 − π o )π u β (1 − ρ)π o π d + ρ
io
n
h
o
u
o d
> 0.
= (1 − β) 1 − β(1 − ρ) 1 − (1 − π )π − π π
χ0 ≡
C.4
h
(82)
A second perturbation of employment
Suppose that in the steady state that the representative family has a positive measure of
unemployed workers who have never been employed. We ask: how would the family’s wealth
change if the set of unemployed workers who have never worked is permanently reduced by
one worker? That is, the family considers sending one such worker to the labor market and,
when he retires, replacing him with an unemployed worker who has never worked. This gives
rise to a stochastic stream of labor income that the family can immediately exchange for
the present value of the stream’s expected value discounted at the stationary interest rate
(1 + i) = β −1 .
We add a twist to this strategy. Whenever the worker (or one of his successors) has
become high-skilled and then loses those skills after an exogenous layoff, the strategy furloughs the worker into unemployment indefinitely and replaces him in the work force with
another unemployed family member who has never worked. This switch of workers yields
a gain to the family because a stream of low unemployment benefits becomes a stream of
high unemployment benefits. The uncertainty associated with retirement makes the gain of
benefits stochastic, but the associated stochastic stream of gains can be sold immediately
ηHw
H
48
for its expected present value, as given by κH
H in expression (62), so that κH = 1−β(1−ρ) .
Let κ00 be the capitalized value of the labor income associated with this strategy of
reducing unemployment among the workers who have never been employed. Moreover, let
κ0H be the capitalized value of the stream of labor income at a future point in time when this
worker (or one of his successors) has attained the high skill level. These capitalized values
48
Recall that newborn workers are also entitled to the lower benefit level ηw.
68
satisfy
κ00
κ0H
o
i
nh
0
o
u 0
o
u
= w + β(1 − ρ) 1 − (1 − π )π κ0 + (1 − π )π κH + βρκ00 ,
n
o
= (1 + H)w + β(1 − ρ) (1 − π o π d )κ0H + π o π d (κ00 + κH
)
+ βρκ00 .
H
(83)
(84)
Equation (84) implies that
κ0H =
h
i
o d
(1 + H)w + β(1 − ρ)π o π d κH
+
β
(1
−
ρ)π
π
+
ρ
κ00
H
1 − β(1 − ρ)(1 − π o π d )
,
(85)
which can be substituted into equation (83),
h
i
h
i
1 − β(1 − ρ)(1 − π o π d ) w + β(1 − ρ)(1 − π o )π u (1 + H)w + β(1 − ρ)π o π d κH
H
κ00 =
0
χ
2
2
o
o u d
β (1 − ρ) (1 − π )π π π ηHw
h
i
,
(86)
= κ4
0 +
1 − β(1 − ρ) χ0
0
where κ4
0 and χ are given by equation (81) and (82), respectively.
C.5
Steady-state consumption
The representative family takes wages and interest rates as given. Since the utility function
is additively separable in consumption and leisure, it is optimal for the family to assign
equal consumption to each of its members. In a steady state with constant consumption,
the stationary interest rate must equal 1 + i = β −1 and the family must be content to hold
a constant level of wealth in the form of physical capital and claims to firms.
Given the family’s optimal labor decisions encoded in flow rates into unemployment
(e0 , e4 ), the representative family has fractions N0 and NH of its members employed with
low skills and high skills, respectively, as determined by equations (73) and (74). The
stationary production of consumption goods per worker c implies per-capita consumption
c̄ = n̄c
(87)
where n̄ is the fraction of employed workers among all members of the family,
n
h
io
1 − (1 − ρ) (1 − π o π d ) − (1 − π o )π u (1 − e0 )ρσ
n̄ = N0 + NH =
.
χe (σ + ρ)
The representative family’s utility in a steady state is
Z 1X
∞
log(c̄) − n̄B
β t u(cjt , njt )dj =
.
1−β
0 t=0
69
(88)
(89)
C.6
Steady-state labor dynamics
It remains to describe optimal labor decisions in a steady state. When the benefit policy
satisfies restriction (64), all high-skilled workers will be employed in a steady state. Unemployment will be confined to workers who currently have low skills. There are two possibilities
concerning steady-state outcomes:
1. e0 = 0 and e4 ∈ [0, 1];
2. e0 ∈ (0, 1] and e4 = 1.
If there is any unemployment among low-skilled workers with low benefits, all high-skilled
workers who suffer skill losses must flow into unemployment, i.e., e4 = 1. If that were not
true, the family would be better off working a low-skilled worker with low benefits instead
of a laid off high-skilled worker who has just suffered a skill loss. Both workers are equally
productive, but the latter is entitled to higher unemployment compensation. Hence, the
steady-state labor dynamics must fall into either class 1 or 2.
What is the optimal setting of the two flow rates into unemployment, (e0 , e4 )? To check
whether a candidate pair of flow rates constitutes a steady state, we consider the welfare
effects of the perturbations to employment that we described above. If the candidate (e0 , e4 )
falls into class 1, we examine the first type of perturbation in which the set of unemployed
workers who have suffered skill loss is permanently reduced by one worker. That increases the
family’s labor income by a capitalized value equal to κ4
0 . In a steady state with equilibrium
−1
gross interest rate β , it would be optimal for the family to convert this capitalized value
into an annuity flow of (1 − β)κ4
0 and permanently to increase consumption by that amount.
The utility derived from this extra flow of consumption should be compared to the loss of
benefits η(1 + H)w and the loss of leisure. The condition for an interior optimum is
h
i
uc (c̄, n̄) (1 − β)κ4
−
η(1
+
H)w
+ un (c̄, n̄) = 0,
(90)
0
where the marginal utilities of consumption and leisure are evaluated at the candidate steadystate allocation. Given our particular utility function, which is additively separable in the
logarithm of consumption and a linear disutility term for labor, the condition for an interior
optimum in class 1 becomes
i
1h
(1 − β)κ4
−
η(1
+
H)w
= B.
(91)
0
c̄
If the candidate (e0 , e4 ) falls into class 2, we consider the second type of perturbation in
which the set of unemployed workers who have never worked is permanently reduced by one
worker. Analyzing this perturbation leads to the following condition for an interior optimum:
h
i
0
uc (c̄, n̄) (1 − β)κ0 − ηw + un (c̄, n̄) = 0,
(92)
which with our preference specification implies
i
1h
(1 − β)κ00 − ηw = B.
c̄
70
(93)
C.7
Employment lotteries
Although the aggregate allocation of labor is unique in an employment lotteries model,
many possible lottery designs that randomly assign different tasks to individual workers can
implement that allocation and yield the same expected utility to workers. In real business
cycle models like the one of Hansen (1985), the identical workers could be randomizing over
an arbitrary number of periods of working and leisure, possibly contingent on the phase of
the business cycle. Alternatively, at the beginning of each period, there could just be an
employment lottery for that period’s labor supply. In our model with ex post heterogeneous
workers, there are two kinds of multiplicity in the design of lotteries. First, the optimal lottery
design is not unique with respect to the identity of low-skilled unemployed workers who are
entitled to low benefits. For example, the workers would be indifferent between the device
proposed above of randomly furloughing newborn workers into life-time unemployment and
other devices that repeatedly randomize employment status among low-skilled workers who
are entitled to low benefits. So long as the devices result in identical aggregate employment
outcomes, workers would derive the same ex ante expected life-time utility. Second, there is
nonuniqueness with respect to the identity of unemployed workers with skill losses whenever
the optimal allocation requires some of these people to work. For example, workers would be
indifferent between the above device of randomly furloughing a fraction of laid off workers
with skill losses into unemployment for the rest of their lives and alternative devices with
higher inflow rates but correspondingly shorter unemployment spells among laid off workers
who experience skill losses.
The equilibrium conditions do restrict the multiplicity of lottery designs in one important
respect. Since the representative family faces no frictions in the labor market and there is a
single wage rate per unit of skill, the family is indifferent between, on the one hand, lotteries
that include only the newly born and laid off old workers and, on the other hand, lotteries
that include all working-age members and that entail furloughing some lottery winners who
are employed into leisure. But firms are not indifferent to these alternative lottery designs
because the latter would result in extra layoff taxes and the loss of firms’ prior investments
in job creation, at least if we assume that productivity processes are lost whenever there are
worker separations, as we have assumed in the case of retirements. It follows that steadystate lottery designs cannot include employed workers because otherwise firms would have the
incentive to offer ‘back-loaded’ wage payments – making the representative family strictly
prefer to assign leisure to the new born and laid off old workers, rather than to furlough
employed workers into leisure.
71
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